Jmat.Real.erfi, branch x greater than or equal to 5

Percentage Accurate: 100.0% → 100.0%

Time: 11.0s

Alternatives: 17

Speedup: 1.9×

• 99.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[x \geq 0.5\]

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\left|x\right|}\\ t_1 := \left(t\_0 \cdot t\_0\right) \cdot t\_0\\ t_2 := \left(t\_1 \cdot t\_0\right) \cdot t\_0\\ \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(t\_0 + \frac{1}{2} \cdot t\_1\right) + \frac{3}{4} \cdot t\_2\right) + \frac{15}{8} \cdot \left(\left(t\_2 \cdot t\_0\right) \cdot t\_0\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (fabs x)))
        (t_1 (* (* t_0 t_0) t_0))
        (t_2 (* (* t_1 t_0) t_0)))
   (*
    (* (/ 1.0 (sqrt PI)) (exp (* (fabs x) (fabs x))))
    (+
     (+ (+ t_0 (* (/ 1.0 2.0) t_1)) (* (/ 3.0 4.0) t_2))
     (* (/ 15.0 8.0) (* (* t_2 t_0) t_0))))))

C

double code(double x) {
	double t_0 = 1.0 / fabs(x);
	double t_1 = (t_0 * t_0) * t_0;
	double t_2 = (t_1 * t_0) * t_0;
	return ((1.0 / sqrt(((double) M_PI))) * exp((fabs(x) * fabs(x)))) * (((t_0 + ((1.0 / 2.0) * t_1)) + ((3.0 / 4.0) * t_2)) + ((15.0 / 8.0) * ((t_2 * t_0) * t_0)));
}

Java

public static double code(double x) {
	double t_0 = 1.0 / Math.abs(x);
	double t_1 = (t_0 * t_0) * t_0;
	double t_2 = (t_1 * t_0) * t_0;
	return ((1.0 / Math.sqrt(Math.PI)) * Math.exp((Math.abs(x) * Math.abs(x)))) * (((t_0 + ((1.0 / 2.0) * t_1)) + ((3.0 / 4.0) * t_2)) + ((15.0 / 8.0) * ((t_2 * t_0) * t_0)));
}

Python

def code(x):
	t_0 = 1.0 / math.fabs(x)
	t_1 = (t_0 * t_0) * t_0
	t_2 = (t_1 * t_0) * t_0
	return ((1.0 / math.sqrt(math.pi)) * math.exp((math.fabs(x) * math.fabs(x)))) * (((t_0 + ((1.0 / 2.0) * t_1)) + ((3.0 / 4.0) * t_2)) + ((15.0 / 8.0) * ((t_2 * t_0) * t_0)))

Julia

function code(x)
	t_0 = Float64(1.0 / abs(x))
	t_1 = Float64(Float64(t_0 * t_0) * t_0)
	t_2 = Float64(Float64(t_1 * t_0) * t_0)
	return Float64(Float64(Float64(1.0 / sqrt(pi)) * exp(Float64(abs(x) * abs(x)))) * Float64(Float64(Float64(t_0 + Float64(Float64(1.0 / 2.0) * t_1)) + Float64(Float64(3.0 / 4.0) * t_2)) + Float64(Float64(15.0 / 8.0) * Float64(Float64(t_2 * t_0) * t_0))))
end

MATLAB

function tmp = code(x)
	t_0 = 1.0 / abs(x);
	t_1 = (t_0 * t_0) * t_0;
	t_2 = (t_1 * t_0) * t_0;
	tmp = ((1.0 / sqrt(pi)) * exp((abs(x) * abs(x)))) * (((t_0 + ((1.0 / 2.0) * t_1)) + ((3.0 / 4.0) * t_2)) + ((15.0 / 8.0) * ((t_2 * t_0) * t_0)));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$1 * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]}, N[(N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(t$95$0 + N[(N[(1.0 / 2.0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(3.0 / 4.0), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] + N[(N[(15.0 / 8.0), $MachinePrecision] * N[(N[(t$95$2 * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\left|x\right|}\\ t_1 := \left(t\_0 \cdot t\_0\right) \cdot t\_0\\ t_2 := \left(t\_1 \cdot t\_0\right) \cdot t\_0\\ \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(t\_0 + \frac{1}{2} \cdot t\_1\right) + \frac{3}{4} \cdot t\_2\right) + \frac{15}{8} \cdot \left(\left(t\_2 \cdot t\_0\right) \cdot t\_0\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (fabs x)))
        (t_1 (* (* t_0 t_0) t_0))
        (t_2 (* (* t_1 t_0) t_0)))
   (*
    (* (/ 1.0 (sqrt PI)) (exp (* (fabs x) (fabs x))))
    (+
     (+ (+ t_0 (* (/ 1.0 2.0) t_1)) (* (/ 3.0 4.0) t_2))
     (* (/ 15.0 8.0) (* (* t_2 t_0) t_0))))))

C

double code(double x) {
	double t_0 = 1.0 / fabs(x);
	double t_1 = (t_0 * t_0) * t_0;
	double t_2 = (t_1 * t_0) * t_0;
	return ((1.0 / sqrt(((double) M_PI))) * exp((fabs(x) * fabs(x)))) * (((t_0 + ((1.0 / 2.0) * t_1)) + ((3.0 / 4.0) * t_2)) + ((15.0 / 8.0) * ((t_2 * t_0) * t_0)));
}

Java

public static double code(double x) {
	double t_0 = 1.0 / Math.abs(x);
	double t_1 = (t_0 * t_0) * t_0;
	double t_2 = (t_1 * t_0) * t_0;
	return ((1.0 / Math.sqrt(Math.PI)) * Math.exp((Math.abs(x) * Math.abs(x)))) * (((t_0 + ((1.0 / 2.0) * t_1)) + ((3.0 / 4.0) * t_2)) + ((15.0 / 8.0) * ((t_2 * t_0) * t_0)));
}

Python

def code(x):
	t_0 = 1.0 / math.fabs(x)
	t_1 = (t_0 * t_0) * t_0
	t_2 = (t_1 * t_0) * t_0
	return ((1.0 / math.sqrt(math.pi)) * math.exp((math.fabs(x) * math.fabs(x)))) * (((t_0 + ((1.0 / 2.0) * t_1)) + ((3.0 / 4.0) * t_2)) + ((15.0 / 8.0) * ((t_2 * t_0) * t_0)))

Julia

function code(x)
	t_0 = Float64(1.0 / abs(x))
	t_1 = Float64(Float64(t_0 * t_0) * t_0)
	t_2 = Float64(Float64(t_1 * t_0) * t_0)
	return Float64(Float64(Float64(1.0 / sqrt(pi)) * exp(Float64(abs(x) * abs(x)))) * Float64(Float64(Float64(t_0 + Float64(Float64(1.0 / 2.0) * t_1)) + Float64(Float64(3.0 / 4.0) * t_2)) + Float64(Float64(15.0 / 8.0) * Float64(Float64(t_2 * t_0) * t_0))))
end

MATLAB

function tmp = code(x)
	t_0 = 1.0 / abs(x);
	t_1 = (t_0 * t_0) * t_0;
	t_2 = (t_1 * t_0) * t_0;
	tmp = ((1.0 / sqrt(pi)) * exp((abs(x) * abs(x)))) * (((t_0 + ((1.0 / 2.0) * t_1)) + ((3.0 / 4.0) * t_2)) + ((15.0 / 8.0) * ((t_2 * t_0) * t_0)));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$1 * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]}, N[(N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(t$95$0 + N[(N[(1.0 / 2.0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(3.0 / 4.0), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] + N[(N[(15.0 / 8.0), $MachinePrecision] * N[(N[(t$95$2 * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Alternative 1: 100.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left|x\right| \cdot x\right)\\ \frac{\left(\left(\frac{1.875}{t\_0 \cdot \left(x \cdot x\right)} + \frac{0.75}{t\_0}\right) + \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right) \cdot {\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (* (* x x) x) (* (fabs x) x))))
   (/
    (*
     (+
      (+ (/ 1.875 (* t_0 (* x x))) (/ 0.75 t_0))
      (/ (+ 1.0 (/ 0.5 (* x x))) (fabs x)))
     (pow (exp x) x))
    (sqrt PI))))

C

double code(double x) {
	double t_0 = ((x * x) * x) * (fabs(x) * x);
	return ((((1.875 / (t_0 * (x * x))) + (0.75 / t_0)) + ((1.0 + (0.5 / (x * x))) / fabs(x))) * pow(exp(x), x)) / sqrt(((double) M_PI));
}

Java

public static double code(double x) {
	double t_0 = ((x * x) * x) * (Math.abs(x) * x);
	return ((((1.875 / (t_0 * (x * x))) + (0.75 / t_0)) + ((1.0 + (0.5 / (x * x))) / Math.abs(x))) * Math.pow(Math.exp(x), x)) / Math.sqrt(Math.PI);
}

Python

def code(x):
	t_0 = ((x * x) * x) * (math.fabs(x) * x)
	return ((((1.875 / (t_0 * (x * x))) + (0.75 / t_0)) + ((1.0 + (0.5 / (x * x))) / math.fabs(x))) * math.pow(math.exp(x), x)) / math.sqrt(math.pi)

Julia

function code(x)
	t_0 = Float64(Float64(Float64(x * x) * x) * Float64(abs(x) * x))
	return Float64(Float64(Float64(Float64(Float64(1.875 / Float64(t_0 * Float64(x * x))) + Float64(0.75 / t_0)) + Float64(Float64(1.0 + Float64(0.5 / Float64(x * x))) / abs(x))) * (exp(x) ^ x)) / sqrt(pi))
end

MATLAB

function tmp = code(x)
	t_0 = ((x * x) * x) * (abs(x) * x);
	tmp = ((((1.875 / (t_0 * (x * x))) + (0.75 / t_0)) + ((1.0 + (0.5 / (x * x))) / abs(x))) * (exp(x) ^ x)) / sqrt(pi);
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * N[(N[Abs[x], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(N[(1.875 / N[(t$95$0 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.75 / t$95$0), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 + N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 100.0%

   \[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]
2. Add Preprocessing

4. Applied rewrites 100.0%

   \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \color{blue}{\left(\frac{1}{\left|x\right|} + \left(\frac{0.5}{\left(x \cdot x\right) \cdot \left|x\right|} + \left(\frac{1.875}{\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left|x\right|\right) \cdot \left(x \cdot x\right)} + \frac{0.75}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left|x\right|}\right)\right)\right)} \]

6. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\frac{e^{x \cdot x} \cdot \left(\frac{\frac{0.5}{x \cdot x} + 1}{\left|x\right|} + \left(\frac{0.75}{\left(\left|x\right| \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)} + \frac{1.875}{\left(\left(\left|x\right| \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(x \cdot x\right)}\right)\right)}{\sqrt{\pi}}} \]
7. Step-by-step derivation

   1. lift-exp.f64 N/A

      \[\leadsto \frac{\color{blue}{e^{x \cdot x}} \cdot \left(\frac{\frac{\frac{1}{2}}{x \cdot x} + 1}{\left|x\right|} + \left(\frac{\frac{3}{4}}{\left(\left|x\right| \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)} + \frac{\frac{15}{8}}{\left(\left(\left|x\right| \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(x \cdot x\right)}\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{e^{\color{blue}{x \cdot x}} \cdot \left(\frac{\frac{\frac{1}{2}}{x \cdot x} + 1}{\left|x\right|} + \left(\frac{\frac{3}{4}}{\left(\left|x\right| \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)} + \frac{\frac{15}{8}}{\left(\left(\left|x\right| \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(x \cdot x\right)}\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}} \]

   3. exp-prod N/A

      \[\leadsto \frac{\color{blue}{{\left(e^{x}\right)}^{x}} \cdot \left(\frac{\frac{\frac{1}{2}}{x \cdot x} + 1}{\left|x\right|} + \left(\frac{\frac{3}{4}}{\left(\left|x\right| \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)} + \frac{\frac{15}{8}}{\left(\left(\left|x\right| \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(x \cdot x\right)}\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}} \]

   4. lower-pow.f64 N/A

      \[\leadsto \frac{\color{blue}{{\left(e^{x}\right)}^{x}} \cdot \left(\frac{\frac{\frac{1}{2}}{x \cdot x} + 1}{\left|x\right|} + \left(\frac{\frac{3}{4}}{\left(\left|x\right| \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)} + \frac{\frac{15}{8}}{\left(\left(\left|x\right| \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(x \cdot x\right)}\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}} \]

   5. lower-exp.f64 100.0

      \[\leadsto \frac{{\color{blue}{\left(e^{x}\right)}}^{x} \cdot \left(\frac{\frac{0.5}{x \cdot x} + 1}{\left|x\right|} + \left(\frac{0.75}{\left(\left|x\right| \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)} + \frac{1.875}{\left(\left(\left|x\right| \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(x \cdot x\right)}\right)\right)}{\sqrt{\pi}} \]

8. Applied rewrites 100.0%

   \[\leadsto \frac{\color{blue}{{\left(e^{x}\right)}^{x}} \cdot \left(\frac{\frac{0.5}{x \cdot x} + 1}{\left|x\right|} + \left(\frac{0.75}{\left(\left|x\right| \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)} + \frac{1.875}{\left(\left(\left|x\right| \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(x \cdot x\right)}\right)\right)}{\sqrt{\pi}} \]

9. Final simplification 100.0%

   \[\leadsto \frac{\left(\left(\frac{1.875}{\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left|x\right| \cdot x\right)\right) \cdot \left(x \cdot x\right)} + \frac{0.75}{\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left|x\right| \cdot x\right)}\right) + \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right) \cdot {\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]
10. Add Preprocessing

Alternative 2: 100.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left|x\right|\\ \left(\left(\left(\frac{0.75}{t\_0} + \frac{1.875}{t\_0 \cdot \left(x \cdot x\right)}\right) + \frac{0.5}{\left|x\right| \cdot \left(x \cdot x\right)}\right) + \frac{1}{\left|x\right|}\right) \cdot \frac{e^{x \cdot x}}{\sqrt{\pi}} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (* (* x x) (* x x)) (fabs x))))
   (*
    (+
     (+
      (+ (/ 0.75 t_0) (/ 1.875 (* t_0 (* x x))))
      (/ 0.5 (* (fabs x) (* x x))))
     (/ 1.0 (fabs x)))
    (/ (exp (* x x)) (sqrt PI)))))

C

double code(double x) {
	double t_0 = ((x * x) * (x * x)) * fabs(x);
	return ((((0.75 / t_0) + (1.875 / (t_0 * (x * x)))) + (0.5 / (fabs(x) * (x * x)))) + (1.0 / fabs(x))) * (exp((x * x)) / sqrt(((double) M_PI)));
}

Java

public static double code(double x) {
	double t_0 = ((x * x) * (x * x)) * Math.abs(x);
	return ((((0.75 / t_0) + (1.875 / (t_0 * (x * x)))) + (0.5 / (Math.abs(x) * (x * x)))) + (1.0 / Math.abs(x))) * (Math.exp((x * x)) / Math.sqrt(Math.PI));
}

Python

def code(x):
	t_0 = ((x * x) * (x * x)) * math.fabs(x)
	return ((((0.75 / t_0) + (1.875 / (t_0 * (x * x)))) + (0.5 / (math.fabs(x) * (x * x)))) + (1.0 / math.fabs(x))) * (math.exp((x * x)) / math.sqrt(math.pi))

Julia

function code(x)
	t_0 = Float64(Float64(Float64(x * x) * Float64(x * x)) * abs(x))
	return Float64(Float64(Float64(Float64(Float64(0.75 / t_0) + Float64(1.875 / Float64(t_0 * Float64(x * x)))) + Float64(0.5 / Float64(abs(x) * Float64(x * x)))) + Float64(1.0 / abs(x))) * Float64(exp(Float64(x * x)) / sqrt(pi)))
end

MATLAB

function tmp = code(x)
	t_0 = ((x * x) * (x * x)) * abs(x);
	tmp = ((((0.75 / t_0) + (1.875 / (t_0 * (x * x)))) + (0.5 / (abs(x) * (x * x)))) + (1.0 / abs(x))) * (exp((x * x)) / sqrt(pi));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(N[(0.75 / t$95$0), $MachinePrecision] + N[(1.875 / N[(t$95$0 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 / N[(N[Abs[x], $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 100.0%

   \[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]
2. Add Preprocessing

4. Applied rewrites 100.0%

   \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \color{blue}{\left(\frac{1}{\left|x\right|} + \left(\frac{0.5}{\left(x \cdot x\right) \cdot \left|x\right|} + \left(\frac{1.875}{\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left|x\right|\right) \cdot \left(x \cdot x\right)} + \frac{0.75}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left|x\right|}\right)\right)\right)} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right)} \cdot \left(\frac{1}{\left|x\right|} + \left(\frac{\frac{1}{2}}{\left(x \cdot x\right) \cdot \left|x\right|} + \left(\frac{\frac{15}{8}}{\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left|x\right|\right) \cdot \left(x \cdot x\right)} + \frac{\frac{3}{4}}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left|x\right|}\right)\right)\right) \]

   2. lift-/.f64 N/A

      \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{\mathsf{PI}\left(\right)}}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\frac{1}{\left|x\right|} + \left(\frac{\frac{1}{2}}{\left(x \cdot x\right) \cdot \left|x\right|} + \left(\frac{\frac{15}{8}}{\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left|x\right|\right) \cdot \left(x \cdot x\right)} + \frac{\frac{3}{4}}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left|x\right|}\right)\right)\right) \]

   3. associate-*l/ N/A

      \[\leadsto \color{blue}{\frac{1 \cdot e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\mathsf{PI}\left(\right)}}} \cdot \left(\frac{1}{\left|x\right|} + \left(\frac{\frac{1}{2}}{\left(x \cdot x\right) \cdot \left|x\right|} + \left(\frac{\frac{15}{8}}{\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left|x\right|\right) \cdot \left(x \cdot x\right)} + \frac{\frac{3}{4}}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left|x\right|}\right)\right)\right) \]

   4. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1 \cdot e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\mathsf{PI}\left(\right)}}} \cdot \left(\frac{1}{\left|x\right|} + \left(\frac{\frac{1}{2}}{\left(x \cdot x\right) \cdot \left|x\right|} + \left(\frac{\frac{15}{8}}{\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left|x\right|\right) \cdot \left(x \cdot x\right)} + \frac{\frac{3}{4}}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left|x\right|}\right)\right)\right) \]

   5. *-lft-identity 100.0

      \[\leadsto \frac{\color{blue}{e^{\left|x\right| \cdot \left|x\right|}}}{\sqrt{\pi}} \cdot \left(\frac{1}{\left|x\right|} + \left(\frac{0.5}{\left(x \cdot x\right) \cdot \left|x\right|} + \left(\frac{1.875}{\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left|x\right|\right) \cdot \left(x \cdot x\right)} + \frac{0.75}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left|x\right|}\right)\right)\right) \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{e^{\color{blue}{\left|x\right| \cdot \left|x\right|}}}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\frac{1}{\left|x\right|} + \left(\frac{\frac{1}{2}}{\left(x \cdot x\right) \cdot \left|x\right|} + \left(\frac{\frac{15}{8}}{\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left|x\right|\right) \cdot \left(x \cdot x\right)} + \frac{\frac{3}{4}}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left|x\right|}\right)\right)\right) \]

   7. lift-fabs.f64 N/A

      \[\leadsto \frac{e^{\color{blue}{\left|x\right|} \cdot \left|x\right|}}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\frac{1}{\left|x\right|} + \left(\frac{\frac{1}{2}}{\left(x \cdot x\right) \cdot \left|x\right|} + \left(\frac{\frac{15}{8}}{\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left|x\right|\right) \cdot \left(x \cdot x\right)} + \frac{\frac{3}{4}}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left|x\right|}\right)\right)\right) \]

   8. lift-fabs.f64 N/A

      \[\leadsto \frac{e^{\left|x\right| \cdot \color{blue}{\left|x\right|}}}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\frac{1}{\left|x\right|} + \left(\frac{\frac{1}{2}}{\left(x \cdot x\right) \cdot \left|x\right|} + \left(\frac{\frac{15}{8}}{\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left|x\right|\right) \cdot \left(x \cdot x\right)} + \frac{\frac{3}{4}}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left|x\right|}\right)\right)\right) \]

   9. sqr-abs N/A

      \[\leadsto \frac{e^{\color{blue}{x \cdot x}}}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\frac{1}{\left|x\right|} + \left(\frac{\frac{1}{2}}{\left(x \cdot x\right) \cdot \left|x\right|} + \left(\frac{\frac{15}{8}}{\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left|x\right|\right) \cdot \left(x \cdot x\right)} + \frac{\frac{3}{4}}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left|x\right|}\right)\right)\right) \]

   10. lift-*.f64 100.0

       \[\leadsto \frac{e^{\color{blue}{x \cdot x}}}{\sqrt{\pi}} \cdot \left(\frac{1}{\left|x\right|} + \left(\frac{0.5}{\left(x \cdot x\right) \cdot \left|x\right|} + \left(\frac{1.875}{\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left|x\right|\right) \cdot \left(x \cdot x\right)} + \frac{0.75}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left|x\right|}\right)\right)\right) \]

6. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\frac{e^{x \cdot x}}{\sqrt{\pi}}} \cdot \left(\frac{1}{\left|x\right|} + \left(\frac{0.5}{\left(x \cdot x\right) \cdot \left|x\right|} + \left(\frac{1.875}{\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left|x\right|\right) \cdot \left(x \cdot x\right)} + \frac{0.75}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left|x\right|}\right)\right)\right) \]

7. Final simplification 100.0%

   \[\leadsto \left(\left(\left(\frac{0.75}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left|x\right|} + \frac{1.875}{\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left|x\right|\right) \cdot \left(x \cdot x\right)}\right) + \frac{0.5}{\left|x\right| \cdot \left(x \cdot x\right)}\right) + \frac{1}{\left|x\right|}\right) \cdot \frac{e^{x \cdot x}}{\sqrt{\pi}} \]
8. Add Preprocessing

Alternative 3: 100.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(x \cdot x\right) \cdot x\right) \cdot x\\ \frac{\left(\frac{1.875}{\left(t\_0 \cdot x\right) \cdot \left(\left|x\right| \cdot x\right)} + \frac{0.75}{t\_0 \cdot \left|x\right|}\right) + \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}}{\sqrt{\pi}} \cdot e^{x \cdot x} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (* (* x x) x) x)))
   (*
    (/
     (+
      (+ (/ 1.875 (* (* t_0 x) (* (fabs x) x))) (/ 0.75 (* t_0 (fabs x))))
      (/ (+ 1.0 (/ 0.5 (* x x))) (fabs x)))
     (sqrt PI))
    (exp (* x x)))))

C

double code(double x) {
	double t_0 = ((x * x) * x) * x;
	return ((((1.875 / ((t_0 * x) * (fabs(x) * x))) + (0.75 / (t_0 * fabs(x)))) + ((1.0 + (0.5 / (x * x))) / fabs(x))) / sqrt(((double) M_PI))) * exp((x * x));
}

Java

public static double code(double x) {
	double t_0 = ((x * x) * x) * x;
	return ((((1.875 / ((t_0 * x) * (Math.abs(x) * x))) + (0.75 / (t_0 * Math.abs(x)))) + ((1.0 + (0.5 / (x * x))) / Math.abs(x))) / Math.sqrt(Math.PI)) * Math.exp((x * x));
}

Python

def code(x):
	t_0 = ((x * x) * x) * x
	return ((((1.875 / ((t_0 * x) * (math.fabs(x) * x))) + (0.75 / (t_0 * math.fabs(x)))) + ((1.0 + (0.5 / (x * x))) / math.fabs(x))) / math.sqrt(math.pi)) * math.exp((x * x))

Julia

function code(x)
	t_0 = Float64(Float64(Float64(x * x) * x) * x)
	return Float64(Float64(Float64(Float64(Float64(1.875 / Float64(Float64(t_0 * x) * Float64(abs(x) * x))) + Float64(0.75 / Float64(t_0 * abs(x)))) + Float64(Float64(1.0 + Float64(0.5 / Float64(x * x))) / abs(x))) / sqrt(pi)) * exp(Float64(x * x)))
end

MATLAB

function tmp = code(x)
	t_0 = ((x * x) * x) * x;
	tmp = ((((1.875 / ((t_0 * x) * (abs(x) * x))) + (0.75 / (t_0 * abs(x)))) + ((1.0 + (0.5 / (x * x))) / abs(x))) / sqrt(pi)) * exp((x * x));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]}, N[(N[(N[(N[(N[(1.875 / N[(N[(t$95$0 * x), $MachinePrecision] * N[(N[Abs[x], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.75 / N[(t$95$0 * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 + N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 100.0%

   \[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]
2. Add Preprocessing

4. Applied rewrites 100.0%

   \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \color{blue}{\left(\frac{1}{\left|x\right|} + \left(\frac{0.5}{\left(x \cdot x\right) \cdot \left|x\right|} + \left(\frac{1.875}{\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left|x\right|\right) \cdot \left(x \cdot x\right)} + \frac{0.75}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left|x\right|}\right)\right)\right)} \]

6. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\frac{e^{x \cdot x} \cdot \left(\frac{\frac{0.5}{x \cdot x} + 1}{\left|x\right|} + \left(\frac{0.75}{\left(\left|x\right| \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)} + \frac{1.875}{\left(\left(\left|x\right| \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(x \cdot x\right)}\right)\right)}{\sqrt{\pi}}} \]

8. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\frac{\frac{\frac{0.5}{x \cdot x} + 1}{\left|x\right|} + \left(\frac{0.75}{\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot \left|x\right|} + \frac{1.875}{\left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot \left(\left|x\right| \cdot x\right)}\right)}{\sqrt{\pi}} \cdot e^{x \cdot x}} \]

9. Final simplification 100.0%

   \[\leadsto \frac{\left(\frac{1.875}{\left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot \left(\left|x\right| \cdot x\right)} + \frac{0.75}{\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot \left|x\right|}\right) + \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}}{\sqrt{\pi}} \cdot e^{x \cdot x} \]
10. Add Preprocessing

Alternative 4: 99.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|x\right| \cdot x\\ \left(\left(\frac{0.5}{t\_0 \cdot x} + \frac{1}{\left|x\right|}\right) + \frac{0.75}{\left(\left(x \cdot x\right) \cdot x\right) \cdot t\_0}\right) \cdot \left(e^{\frac{1}{\frac{1}{x \cdot x}}} \cdot \frac{1}{\sqrt{\pi}}\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (fabs x) x)))
   (*
    (+ (+ (/ 0.5 (* t_0 x)) (/ 1.0 (fabs x))) (/ 0.75 (* (* (* x x) x) t_0)))
    (* (exp (/ 1.0 (/ 1.0 (* x x)))) (/ 1.0 (sqrt PI))))))

C

double code(double x) {
	double t_0 = fabs(x) * x;
	return (((0.5 / (t_0 * x)) + (1.0 / fabs(x))) + (0.75 / (((x * x) * x) * t_0))) * (exp((1.0 / (1.0 / (x * x)))) * (1.0 / sqrt(((double) M_PI))));
}

Java

public static double code(double x) {
	double t_0 = Math.abs(x) * x;
	return (((0.5 / (t_0 * x)) + (1.0 / Math.abs(x))) + (0.75 / (((x * x) * x) * t_0))) * (Math.exp((1.0 / (1.0 / (x * x)))) * (1.0 / Math.sqrt(Math.PI)));
}

Python

def code(x):
	t_0 = math.fabs(x) * x
	return (((0.5 / (t_0 * x)) + (1.0 / math.fabs(x))) + (0.75 / (((x * x) * x) * t_0))) * (math.exp((1.0 / (1.0 / (x * x)))) * (1.0 / math.sqrt(math.pi)))

Julia

function code(x)
	t_0 = Float64(abs(x) * x)
	return Float64(Float64(Float64(Float64(0.5 / Float64(t_0 * x)) + Float64(1.0 / abs(x))) + Float64(0.75 / Float64(Float64(Float64(x * x) * x) * t_0))) * Float64(exp(Float64(1.0 / Float64(1.0 / Float64(x * x)))) * Float64(1.0 / sqrt(pi))))
end

MATLAB

function tmp = code(x)
	t_0 = abs(x) * x;
	tmp = (((0.5 / (t_0 * x)) + (1.0 / abs(x))) + (0.75 / (((x * x) * x) * t_0))) * (exp((1.0 / (1.0 / (x * x)))) * (1.0 / sqrt(pi)));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Abs[x], $MachinePrecision] * x), $MachinePrecision]}, N[(N[(N[(N[(0.5 / N[(t$95$0 * x), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.75 / N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(1.0 / N[(1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 100.0%

   \[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]
2. Add Preprocessing

4. Applied rewrites 100.0%

   \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \color{blue}{\left(\frac{1}{\left|x\right|} + \left(\frac{0.5}{\left(x \cdot x\right) \cdot \left|x\right|} + \left(\frac{1.875}{\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left|x\right|\right) \cdot \left(x \cdot x\right)} + \frac{0.75}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left|x\right|}\right)\right)\right)} \]

5. Taylor expanded in x around inf

   \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \color{blue}{\left(\frac{\frac{1}{2}}{{x}^{2} \cdot \left|x\right|} + \left(\frac{\frac{3}{4}}{{x}^{4} \cdot \left|x\right|} + \frac{1}{\left|x\right|}\right)\right)} \]

7. Applied rewrites 99.0%

   \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \color{blue}{\left(\left(\frac{0.5}{\left(\left|x\right| \cdot x\right) \cdot x} + \frac{1}{\left|x\right|}\right) + \frac{0.75}{\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left|x\right| \cdot x\right)}\right)} \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\color{blue}{\left|x\right| \cdot \left|x\right|}}\right) \cdot \left(\left(\frac{\frac{1}{2}}{\left(\left|x\right| \cdot x\right) \cdot x} + \frac{1}{\left|x\right|}\right) + \frac{\frac{3}{4}}{\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left|x\right| \cdot x\right)}\right) \]

   2. lift-fabs.f64 N/A

      \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\color{blue}{\left|x\right|} \cdot \left|x\right|}\right) \cdot \left(\left(\frac{\frac{1}{2}}{\left(\left|x\right| \cdot x\right) \cdot x} + \frac{1}{\left|x\right|}\right) + \frac{\frac{3}{4}}{\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left|x\right| \cdot x\right)}\right) \]

   3. lift-fabs.f64 N/A

      \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \color{blue}{\left|x\right|}}\right) \cdot \left(\left(\frac{\frac{1}{2}}{\left(\left|x\right| \cdot x\right) \cdot x} + \frac{1}{\left|x\right|}\right) + \frac{\frac{3}{4}}{\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left|x\right| \cdot x\right)}\right) \]

   4. sqr-abs N/A

      \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\color{blue}{x \cdot x}}\right) \cdot \left(\left(\frac{\frac{1}{2}}{\left(\left|x\right| \cdot x\right) \cdot x} + \frac{1}{\left|x\right|}\right) + \frac{\frac{3}{4}}{\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left|x\right| \cdot x\right)}\right) \]

   5. lift-*.f64 99.0

      \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\color{blue}{x \cdot x}}\right) \cdot \left(\left(\frac{0.5}{\left(\left|x\right| \cdot x\right) \cdot x} + \frac{1}{\left|x\right|}\right) + \frac{0.75}{\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left|x\right| \cdot x\right)}\right) \]

   6. /-rgt-identity N/A

      \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\color{blue}{\frac{x \cdot x}{1}}}\right) \cdot \left(\left(\frac{\frac{1}{2}}{\left(\left|x\right| \cdot x\right) \cdot x} + \frac{1}{\left|x\right|}\right) + \frac{\frac{3}{4}}{\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left|x\right| \cdot x\right)}\right) \]

   7. clear-num N/A

      \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\color{blue}{\frac{1}{\frac{1}{x \cdot x}}}}\right) \cdot \left(\left(\frac{\frac{1}{2}}{\left(\left|x\right| \cdot x\right) \cdot x} + \frac{1}{\left|x\right|}\right) + \frac{\frac{3}{4}}{\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left|x\right| \cdot x\right)}\right) \]

   8. lift-*.f64 N/A

      \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\frac{1}{\frac{1}{\color{blue}{x \cdot x}}}}\right) \cdot \left(\left(\frac{\frac{1}{2}}{\left(\left|x\right| \cdot x\right) \cdot x} + \frac{1}{\left|x\right|}\right) + \frac{\frac{3}{4}}{\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left|x\right| \cdot x\right)}\right) \]

   9. lower-/.f64 N/A

      \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\color{blue}{\frac{1}{\frac{1}{x \cdot x}}}}\right) \cdot \left(\left(\frac{\frac{1}{2}}{\left(\left|x\right| \cdot x\right) \cdot x} + \frac{1}{\left|x\right|}\right) + \frac{\frac{3}{4}}{\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left|x\right| \cdot x\right)}\right) \]

   10. lift-*.f64 N/A

       \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\frac{1}{\frac{1}{\color{blue}{x \cdot x}}}}\right) \cdot \left(\left(\frac{\frac{1}{2}}{\left(\left|x\right| \cdot x\right) \cdot x} + \frac{1}{\left|x\right|}\right) + \frac{\frac{3}{4}}{\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left|x\right| \cdot x\right)}\right) \]

   11. lower-/.f64 99.0

       \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\frac{1}{\color{blue}{\frac{1}{x \cdot x}}}}\right) \cdot \left(\left(\frac{0.5}{\left(\left|x\right| \cdot x\right) \cdot x} + \frac{1}{\left|x\right|}\right) + \frac{0.75}{\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left|x\right| \cdot x\right)}\right) \]

9. Applied rewrites 99.0%

   \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\color{blue}{\frac{1}{\frac{1}{x \cdot x}}}}\right) \cdot \left(\left(\frac{0.5}{\left(\left|x\right| \cdot x\right) \cdot x} + \frac{1}{\left|x\right|}\right) + \frac{0.75}{\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left|x\right| \cdot x\right)}\right) \]

10. Final simplification 99.0%

    \[\leadsto \left(\left(\frac{0.5}{\left(\left|x\right| \cdot x\right) \cdot x} + \frac{1}{\left|x\right|}\right) + \frac{0.75}{\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left|x\right| \cdot x\right)}\right) \cdot \left(e^{\frac{1}{\frac{1}{x \cdot x}}} \cdot \frac{1}{\sqrt{\pi}}\right) \]
11. Add Preprocessing

Alternative 5: 99.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|x\right| \cdot x\\ \frac{1}{\frac{\sqrt{\pi}}{e^{x \cdot x}}} \cdot \left(\left(\frac{0.5}{t\_0 \cdot x} + \frac{1}{\left|x\right|}\right) + \frac{0.75}{\left(\left(x \cdot x\right) \cdot x\right) \cdot t\_0}\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (fabs x) x)))
   (*
    (/ 1.0 (/ (sqrt PI) (exp (* x x))))
    (+
     (+ (/ 0.5 (* t_0 x)) (/ 1.0 (fabs x)))
     (/ 0.75 (* (* (* x x) x) t_0))))))

C

double code(double x) {
	double t_0 = fabs(x) * x;
	return (1.0 / (sqrt(((double) M_PI)) / exp((x * x)))) * (((0.5 / (t_0 * x)) + (1.0 / fabs(x))) + (0.75 / (((x * x) * x) * t_0)));
}

Java

public static double code(double x) {
	double t_0 = Math.abs(x) * x;
	return (1.0 / (Math.sqrt(Math.PI) / Math.exp((x * x)))) * (((0.5 / (t_0 * x)) + (1.0 / Math.abs(x))) + (0.75 / (((x * x) * x) * t_0)));
}

Python

def code(x):
	t_0 = math.fabs(x) * x
	return (1.0 / (math.sqrt(math.pi) / math.exp((x * x)))) * (((0.5 / (t_0 * x)) + (1.0 / math.fabs(x))) + (0.75 / (((x * x) * x) * t_0)))

Julia

function code(x)
	t_0 = Float64(abs(x) * x)
	return Float64(Float64(1.0 / Float64(sqrt(pi) / exp(Float64(x * x)))) * Float64(Float64(Float64(0.5 / Float64(t_0 * x)) + Float64(1.0 / abs(x))) + Float64(0.75 / Float64(Float64(Float64(x * x) * x) * t_0))))
end

MATLAB

function tmp = code(x)
	t_0 = abs(x) * x;
	tmp = (1.0 / (sqrt(pi) / exp((x * x)))) * (((0.5 / (t_0 * x)) + (1.0 / abs(x))) + (0.75 / (((x * x) * x) * t_0)));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Abs[x], $MachinePrecision] * x), $MachinePrecision]}, N[(N[(1.0 / N[(N[Sqrt[Pi], $MachinePrecision] / N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(0.5 / N[(t$95$0 * x), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.75 / N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 100.0%

   \[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]
2. Add Preprocessing

4. Applied rewrites 100.0%

   \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \color{blue}{\left(\frac{1}{\left|x\right|} + \left(\frac{0.5}{\left(x \cdot x\right) \cdot \left|x\right|} + \left(\frac{1.875}{\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left|x\right|\right) \cdot \left(x \cdot x\right)} + \frac{0.75}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left|x\right|}\right)\right)\right)} \]

5. Taylor expanded in x around inf

   \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \color{blue}{\left(\frac{\frac{1}{2}}{{x}^{2} \cdot \left|x\right|} + \left(\frac{\frac{3}{4}}{{x}^{4} \cdot \left|x\right|} + \frac{1}{\left|x\right|}\right)\right)} \]

7. Applied rewrites 99.0%

   \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \color{blue}{\left(\left(\frac{0.5}{\left(\left|x\right| \cdot x\right) \cdot x} + \frac{1}{\left|x\right|}\right) + \frac{0.75}{\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left|x\right| \cdot x\right)}\right)} \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right)} \cdot \left(\left(\frac{\frac{1}{2}}{\left(\left|x\right| \cdot x\right) \cdot x} + \frac{1}{\left|x\right|}\right) + \frac{\frac{3}{4}}{\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left|x\right| \cdot x\right)}\right) \]

   2. lift-/.f64 N/A

      \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{\mathsf{PI}\left(\right)}}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\frac{\frac{1}{2}}{\left(\left|x\right| \cdot x\right) \cdot x} + \frac{1}{\left|x\right|}\right) + \frac{\frac{3}{4}}{\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left|x\right| \cdot x\right)}\right) \]

   3. lift-*.f64 N/A

      \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\color{blue}{\left|x\right| \cdot \left|x\right|}}\right) \cdot \left(\left(\frac{\frac{1}{2}}{\left(\left|x\right| \cdot x\right) \cdot x} + \frac{1}{\left|x\right|}\right) + \frac{\frac{3}{4}}{\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left|x\right| \cdot x\right)}\right) \]

   4. lift-fabs.f64 N/A

      \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\color{blue}{\left|x\right|} \cdot \left|x\right|}\right) \cdot \left(\left(\frac{\frac{1}{2}}{\left(\left|x\right| \cdot x\right) \cdot x} + \frac{1}{\left|x\right|}\right) + \frac{\frac{3}{4}}{\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left|x\right| \cdot x\right)}\right) \]

   5. lift-fabs.f64 N/A

      \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \color{blue}{\left|x\right|}}\right) \cdot \left(\left(\frac{\frac{1}{2}}{\left(\left|x\right| \cdot x\right) \cdot x} + \frac{1}{\left|x\right|}\right) + \frac{\frac{3}{4}}{\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left|x\right| \cdot x\right)}\right) \]

   6. sqr-abs N/A

      \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\color{blue}{x \cdot x}}\right) \cdot \left(\left(\frac{\frac{1}{2}}{\left(\left|x\right| \cdot x\right) \cdot x} + \frac{1}{\left|x\right|}\right) + \frac{\frac{3}{4}}{\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left|x\right| \cdot x\right)}\right) \]

   7. lift-*.f64 N/A

      \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\color{blue}{x \cdot x}}\right) \cdot \left(\left(\frac{\frac{1}{2}}{\left(\left|x\right| \cdot x\right) \cdot x} + \frac{1}{\left|x\right|}\right) + \frac{\frac{3}{4}}{\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left|x\right| \cdot x\right)}\right) \]

   8. associate-/r/ N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{\mathsf{PI}\left(\right)}}{e^{x \cdot x}}}} \cdot \left(\left(\frac{\frac{1}{2}}{\left(\left|x\right| \cdot x\right) \cdot x} + \frac{1}{\left|x\right|}\right) + \frac{\frac{3}{4}}{\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left|x\right| \cdot x\right)}\right) \]

   9. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{\mathsf{PI}\left(\right)}}{e^{x \cdot x}}}} \cdot \left(\left(\frac{\frac{1}{2}}{\left(\left|x\right| \cdot x\right) \cdot x} + \frac{1}{\left|x\right|}\right) + \frac{\frac{3}{4}}{\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left|x\right| \cdot x\right)}\right) \]

   10. lower-/.f64 99.0

       \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{\pi}}{e^{x \cdot x}}}} \cdot \left(\left(\frac{0.5}{\left(\left|x\right| \cdot x\right) \cdot x} + \frac{1}{\left|x\right|}\right) + \frac{0.75}{\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left|x\right| \cdot x\right)}\right) \]

9. Applied rewrites 99.0%

   \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{\pi}}{e^{x \cdot x}}}} \cdot \left(\left(\frac{0.5}{\left(\left|x\right| \cdot x\right) \cdot x} + \frac{1}{\left|x\right|}\right) + \frac{0.75}{\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left|x\right| \cdot x\right)}\right) \]
10. Add Preprocessing

Alternative 6: 99.7% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \left(\left(\frac{0.5}{\left(\left|x\right| \cdot x\right) \cdot x} + \frac{1}{\left|x\right|}\right) + \frac{0.75}{\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot \left|x\right|}\right) \cdot \frac{e^{x \cdot x}}{\sqrt{\pi}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (+
   (+ (/ 0.5 (* (* (fabs x) x) x)) (/ 1.0 (fabs x)))
   (/ 0.75 (* (* (* (* x x) x) x) (fabs x))))
  (/ (exp (* x x)) (sqrt PI))))

C

double code(double x) {
	return (((0.5 / ((fabs(x) * x) * x)) + (1.0 / fabs(x))) + (0.75 / ((((x * x) * x) * x) * fabs(x)))) * (exp((x * x)) / sqrt(((double) M_PI)));
}

Java

public static double code(double x) {
	return (((0.5 / ((Math.abs(x) * x) * x)) + (1.0 / Math.abs(x))) + (0.75 / ((((x * x) * x) * x) * Math.abs(x)))) * (Math.exp((x * x)) / Math.sqrt(Math.PI));
}

Python

def code(x):
	return (((0.5 / ((math.fabs(x) * x) * x)) + (1.0 / math.fabs(x))) + (0.75 / ((((x * x) * x) * x) * math.fabs(x)))) * (math.exp((x * x)) / math.sqrt(math.pi))

Julia

function code(x)
	return Float64(Float64(Float64(Float64(0.5 / Float64(Float64(abs(x) * x) * x)) + Float64(1.0 / abs(x))) + Float64(0.75 / Float64(Float64(Float64(Float64(x * x) * x) * x) * abs(x)))) * Float64(exp(Float64(x * x)) / sqrt(pi)))
end

MATLAB

function tmp = code(x)
	tmp = (((0.5 / ((abs(x) * x) * x)) + (1.0 / abs(x))) + (0.75 / ((((x * x) * x) * x) * abs(x)))) * (exp((x * x)) / sqrt(pi));
end

Wolfram

code[x_] := N[(N[(N[(N[(0.5 / N[(N[(N[Abs[x], $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.75 / N[(N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]
2. Add Preprocessing

4. Applied rewrites 100.0%

   \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \color{blue}{\left(\frac{1}{\left|x\right|} + \left(\frac{0.5}{\left(x \cdot x\right) \cdot \left|x\right|} + \left(\frac{1.875}{\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left|x\right|\right) \cdot \left(x \cdot x\right)} + \frac{0.75}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left|x\right|}\right)\right)\right)} \]

5. Taylor expanded in x around inf

   \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \color{blue}{\left(\frac{\frac{1}{2}}{{x}^{2} \cdot \left|x\right|} + \left(\frac{\frac{3}{4}}{{x}^{4} \cdot \left|x\right|} + \frac{1}{\left|x\right|}\right)\right)} \]

7. Applied rewrites 99.0%

   \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \color{blue}{\left(\left(\frac{0.5}{\left(\left|x\right| \cdot x\right) \cdot x} + \frac{1}{\left|x\right|}\right) + \frac{0.75}{\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left|x\right| \cdot x\right)}\right)} \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right)} \cdot \left(\left(\frac{\frac{1}{2}}{\left(\left|x\right| \cdot x\right) \cdot x} + \frac{1}{\left|x\right|}\right) + \frac{\frac{3}{4}}{\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left|x\right| \cdot x\right)}\right) \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\left(e^{\left|x\right| \cdot \left|x\right|} \cdot \frac{1}{\sqrt{\mathsf{PI}\left(\right)}}\right)} \cdot \left(\left(\frac{\frac{1}{2}}{\left(\left|x\right| \cdot x\right) \cdot x} + \frac{1}{\left|x\right|}\right) + \frac{\frac{3}{4}}{\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left|x\right| \cdot x\right)}\right) \]

   3. lift-*.f64 N/A

      \[\leadsto \left(e^{\color{blue}{\left|x\right| \cdot \left|x\right|}} \cdot \frac{1}{\sqrt{\mathsf{PI}\left(\right)}}\right) \cdot \left(\left(\frac{\frac{1}{2}}{\left(\left|x\right| \cdot x\right) \cdot x} + \frac{1}{\left|x\right|}\right) + \frac{\frac{3}{4}}{\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left|x\right| \cdot x\right)}\right) \]

   4. lift-fabs.f64 N/A

      \[\leadsto \left(e^{\color{blue}{\left|x\right|} \cdot \left|x\right|} \cdot \frac{1}{\sqrt{\mathsf{PI}\left(\right)}}\right) \cdot \left(\left(\frac{\frac{1}{2}}{\left(\left|x\right| \cdot x\right) \cdot x} + \frac{1}{\left|x\right|}\right) + \frac{\frac{3}{4}}{\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left|x\right| \cdot x\right)}\right) \]

   5. lift-fabs.f64 N/A

      \[\leadsto \left(e^{\left|x\right| \cdot \color{blue}{\left|x\right|}} \cdot \frac{1}{\sqrt{\mathsf{PI}\left(\right)}}\right) \cdot \left(\left(\frac{\frac{1}{2}}{\left(\left|x\right| \cdot x\right) \cdot x} + \frac{1}{\left|x\right|}\right) + \frac{\frac{3}{4}}{\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left|x\right| \cdot x\right)}\right) \]

   6. sqr-abs N/A

      \[\leadsto \left(e^{\color{blue}{x \cdot x}} \cdot \frac{1}{\sqrt{\mathsf{PI}\left(\right)}}\right) \cdot \left(\left(\frac{\frac{1}{2}}{\left(\left|x\right| \cdot x\right) \cdot x} + \frac{1}{\left|x\right|}\right) + \frac{\frac{3}{4}}{\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left|x\right| \cdot x\right)}\right) \]

   7. lift-*.f64 N/A

      \[\leadsto \left(e^{\color{blue}{x \cdot x}} \cdot \frac{1}{\sqrt{\mathsf{PI}\left(\right)}}\right) \cdot \left(\left(\frac{\frac{1}{2}}{\left(\left|x\right| \cdot x\right) \cdot x} + \frac{1}{\left|x\right|}\right) + \frac{\frac{3}{4}}{\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left|x\right| \cdot x\right)}\right) \]

   8. lift-/.f64 N/A

      \[\leadsto \left(e^{x \cdot x} \cdot \color{blue}{\frac{1}{\sqrt{\mathsf{PI}\left(\right)}}}\right) \cdot \left(\left(\frac{\frac{1}{2}}{\left(\left|x\right| \cdot x\right) \cdot x} + \frac{1}{\left|x\right|}\right) + \frac{\frac{3}{4}}{\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left|x\right| \cdot x\right)}\right) \]

   9. div-inv N/A

      \[\leadsto \color{blue}{\frac{e^{x \cdot x}}{\sqrt{\mathsf{PI}\left(\right)}}} \cdot \left(\left(\frac{\frac{1}{2}}{\left(\left|x\right| \cdot x\right) \cdot x} + \frac{1}{\left|x\right|}\right) + \frac{\frac{3}{4}}{\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left|x\right| \cdot x\right)}\right) \]

   10. lift-/.f64 99.0

       \[\leadsto \color{blue}{\frac{e^{x \cdot x}}{\sqrt{\pi}}} \cdot \left(\left(\frac{0.5}{\left(\left|x\right| \cdot x\right) \cdot x} + \frac{1}{\left|x\right|}\right) + \frac{0.75}{\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left|x\right| \cdot x\right)}\right) \]

9. Applied rewrites 99.0%

   \[\leadsto \color{blue}{\frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(\left(\frac{0.5}{\left(\left|x\right| \cdot x\right) \cdot x} + \frac{1}{\left|x\right|}\right) + \frac{0.75}{\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot \left|x\right|}\right)} \]

10. Final simplification 99.0%

    \[\leadsto \left(\left(\frac{0.5}{\left(\left|x\right| \cdot x\right) \cdot x} + \frac{1}{\left|x\right|}\right) + \frac{0.75}{\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot \left|x\right|}\right) \cdot \frac{e^{x \cdot x}}{\sqrt{\pi}} \]
11. Add Preprocessing

Alternative 7: 99.7% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \frac{\left(\frac{0.75}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)} + \left(1 + \frac{0.5}{x \cdot x}\right)\right) \cdot \frac{e^{x \cdot x}}{\left|x\right|}}{\sqrt{\pi}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  (*
   (+ (/ 0.75 (* (* x x) (* x x))) (+ 1.0 (/ 0.5 (* x x))))
   (/ (exp (* x x)) (fabs x)))
  (sqrt PI)))

C

double code(double x) {
	return (((0.75 / ((x * x) * (x * x))) + (1.0 + (0.5 / (x * x)))) * (exp((x * x)) / fabs(x))) / sqrt(((double) M_PI));
}

Java

public static double code(double x) {
	return (((0.75 / ((x * x) * (x * x))) + (1.0 + (0.5 / (x * x)))) * (Math.exp((x * x)) / Math.abs(x))) / Math.sqrt(Math.PI);
}

Python

def code(x):
	return (((0.75 / ((x * x) * (x * x))) + (1.0 + (0.5 / (x * x)))) * (math.exp((x * x)) / math.fabs(x))) / math.sqrt(math.pi)

Julia

function code(x)
	return Float64(Float64(Float64(Float64(0.75 / Float64(Float64(x * x) * Float64(x * x))) + Float64(1.0 + Float64(0.5 / Float64(x * x)))) * Float64(exp(Float64(x * x)) / abs(x))) / sqrt(pi))
end

MATLAB

function tmp = code(x)
	tmp = (((0.75 / ((x * x) * (x * x))) + (1.0 + (0.5 / (x * x)))) * (exp((x * x)) / abs(x))) / sqrt(pi);
end

Wolfram

code[x_] := N[(N[(N[(N[(0.75 / N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision] / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]
2. Add Preprocessing

4. Applied rewrites 100.0%

   \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \color{blue}{\left(\frac{1}{\left|x\right|} + \left(\frac{0.5}{\left(x \cdot x\right) \cdot \left|x\right|} + \left(\frac{1.875}{\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left|x\right|\right) \cdot \left(x \cdot x\right)} + \frac{0.75}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left|x\right|}\right)\right)\right)} \]

6. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\frac{e^{x \cdot x} \cdot \left(\frac{\frac{0.5}{x \cdot x} + 1}{\left|x\right|} + \left(\frac{0.75}{\left(\left|x\right| \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)} + \frac{1.875}{\left(\left(\left|x\right| \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(x \cdot x\right)}\right)\right)}{\sqrt{\pi}}} \]
7. Step-by-step derivation

   1. lift-exp.f64 N/A

      \[\leadsto \frac{\color{blue}{e^{x \cdot x}} \cdot \left(\frac{\frac{\frac{1}{2}}{x \cdot x} + 1}{\left|x\right|} + \left(\frac{\frac{3}{4}}{\left(\left|x\right| \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)} + \frac{\frac{15}{8}}{\left(\left(\left|x\right| \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(x \cdot x\right)}\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{e^{\color{blue}{x \cdot x}} \cdot \left(\frac{\frac{\frac{1}{2}}{x \cdot x} + 1}{\left|x\right|} + \left(\frac{\frac{3}{4}}{\left(\left|x\right| \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)} + \frac{\frac{15}{8}}{\left(\left(\left|x\right| \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(x \cdot x\right)}\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}} \]

   3. exp-prod N/A

      \[\leadsto \frac{\color{blue}{{\left(e^{x}\right)}^{x}} \cdot \left(\frac{\frac{\frac{1}{2}}{x \cdot x} + 1}{\left|x\right|} + \left(\frac{\frac{3}{4}}{\left(\left|x\right| \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)} + \frac{\frac{15}{8}}{\left(\left(\left|x\right| \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(x \cdot x\right)}\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}} \]

   4. lower-pow.f64 N/A

      \[\leadsto \frac{\color{blue}{{\left(e^{x}\right)}^{x}} \cdot \left(\frac{\frac{\frac{1}{2}}{x \cdot x} + 1}{\left|x\right|} + \left(\frac{\frac{3}{4}}{\left(\left|x\right| \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)} + \frac{\frac{15}{8}}{\left(\left(\left|x\right| \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(x \cdot x\right)}\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}} \]

   5. lower-exp.f64 100.0

      \[\leadsto \frac{{\color{blue}{\left(e^{x}\right)}}^{x} \cdot \left(\frac{\frac{0.5}{x \cdot x} + 1}{\left|x\right|} + \left(\frac{0.75}{\left(\left|x\right| \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)} + \frac{1.875}{\left(\left(\left|x\right| \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(x \cdot x\right)}\right)\right)}{\sqrt{\pi}} \]

8. Applied rewrites 100.0%

   \[\leadsto \frac{\color{blue}{{\left(e^{x}\right)}^{x}} \cdot \left(\frac{\frac{0.5}{x \cdot x} + 1}{\left|x\right|} + \left(\frac{0.75}{\left(\left|x\right| \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)} + \frac{1.875}{\left(\left(\left|x\right| \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(x \cdot x\right)}\right)\right)}{\sqrt{\pi}} \]

9. Taylor expanded in x around inf

   \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot \frac{e^{{x}^{2}}}{{x}^{2} \cdot \left|x\right|} + \left(\frac{3}{4} \cdot \frac{e^{{x}^{2}}}{{x}^{4} \cdot \left|x\right|} + \frac{e^{{x}^{2}}}{\left|x\right|}\right)}}{\sqrt{\mathsf{PI}\left(\right)}} \]
10. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto \frac{\color{blue}{\left(\frac{3}{4} \cdot \frac{e^{{x}^{2}}}{{x}^{4} \cdot \left|x\right|} + \frac{e^{{x}^{2}}}{\left|x\right|}\right) + \frac{1}{2} \cdot \frac{e^{{x}^{2}}}{{x}^{2} \cdot \left|x\right|}}}{\sqrt{\mathsf{PI}\left(\right)}} \]

    2. associate-+l+ N/A

       \[\leadsto \frac{\color{blue}{\frac{3}{4} \cdot \frac{e^{{x}^{2}}}{{x}^{4} \cdot \left|x\right|} + \left(\frac{e^{{x}^{2}}}{\left|x\right|} + \frac{1}{2} \cdot \frac{e^{{x}^{2}}}{{x}^{2} \cdot \left|x\right|}\right)}}{\sqrt{\mathsf{PI}\left(\right)}} \]

    3. associate-*r/ N/A

       \[\leadsto \frac{\color{blue}{\frac{\frac{3}{4} \cdot e^{{x}^{2}}}{{x}^{4} \cdot \left|x\right|}} + \left(\frac{e^{{x}^{2}}}{\left|x\right|} + \frac{1}{2} \cdot \frac{e^{{x}^{2}}}{{x}^{2} \cdot \left|x\right|}\right)}{\sqrt{\mathsf{PI}\left(\right)}} \]

    4. times-frac N/A

       \[\leadsto \frac{\color{blue}{\frac{\frac{3}{4}}{{x}^{4}} \cdot \frac{e^{{x}^{2}}}{\left|x\right|}} + \left(\frac{e^{{x}^{2}}}{\left|x\right|} + \frac{1}{2} \cdot \frac{e^{{x}^{2}}}{{x}^{2} \cdot \left|x\right|}\right)}{\sqrt{\mathsf{PI}\left(\right)}} \]

    5. unpow2 N/A

       \[\leadsto \frac{\frac{\frac{3}{4}}{{x}^{4}} \cdot \frac{e^{\color{blue}{x \cdot x}}}{\left|x\right|} + \left(\frac{e^{{x}^{2}}}{\left|x\right|} + \frac{1}{2} \cdot \frac{e^{{x}^{2}}}{{x}^{2} \cdot \left|x\right|}\right)}{\sqrt{\mathsf{PI}\left(\right)}} \]

    6. sqr-abs N/A

       \[\leadsto \frac{\frac{\frac{3}{4}}{{x}^{4}} \cdot \frac{e^{\color{blue}{\left|x\right| \cdot \left|x\right|}}}{\left|x\right|} + \left(\frac{e^{{x}^{2}}}{\left|x\right|} + \frac{1}{2} \cdot \frac{e^{{x}^{2}}}{{x}^{2} \cdot \left|x\right|}\right)}{\sqrt{\mathsf{PI}\left(\right)}} \]

    7. unpow2 N/A

       \[\leadsto \frac{\frac{\frac{3}{4}}{{x}^{4}} \cdot \frac{e^{\color{blue}{{\left(\left|x\right|\right)}^{2}}}}{\left|x\right|} + \left(\frac{e^{{x}^{2}}}{\left|x\right|} + \frac{1}{2} \cdot \frac{e^{{x}^{2}}}{{x}^{2} \cdot \left|x\right|}\right)}{\sqrt{\mathsf{PI}\left(\right)}} \]

    8. unpow2 N/A

       \[\leadsto \frac{\frac{\frac{3}{4}}{{x}^{4}} \cdot \frac{e^{{\left(\left|x\right|\right)}^{2}}}{\left|x\right|} + \left(\frac{e^{\color{blue}{x \cdot x}}}{\left|x\right|} + \frac{1}{2} \cdot \frac{e^{{x}^{2}}}{{x}^{2} \cdot \left|x\right|}\right)}{\sqrt{\mathsf{PI}\left(\right)}} \]

    9. sqr-abs N/A

       \[\leadsto \frac{\frac{\frac{3}{4}}{{x}^{4}} \cdot \frac{e^{{\left(\left|x\right|\right)}^{2}}}{\left|x\right|} + \left(\frac{e^{\color{blue}{\left|x\right| \cdot \left|x\right|}}}{\left|x\right|} + \frac{1}{2} \cdot \frac{e^{{x}^{2}}}{{x}^{2} \cdot \left|x\right|}\right)}{\sqrt{\mathsf{PI}\left(\right)}} \]

    10. unpow2 N/A

        \[\leadsto \frac{\frac{\frac{3}{4}}{{x}^{4}} \cdot \frac{e^{{\left(\left|x\right|\right)}^{2}}}{\left|x\right|} + \left(\frac{e^{\color{blue}{{\left(\left|x\right|\right)}^{2}}}}{\left|x\right|} + \frac{1}{2} \cdot \frac{e^{{x}^{2}}}{{x}^{2} \cdot \left|x\right|}\right)}{\sqrt{\mathsf{PI}\left(\right)}} \]

    11. associate-*r/ N/A

        \[\leadsto \frac{\frac{\frac{3}{4}}{{x}^{4}} \cdot \frac{e^{{\left(\left|x\right|\right)}^{2}}}{\left|x\right|} + \left(\frac{e^{{\left(\left|x\right|\right)}^{2}}}{\left|x\right|} + \color{blue}{\frac{\frac{1}{2} \cdot e^{{x}^{2}}}{{x}^{2} \cdot \left|x\right|}}\right)}{\sqrt{\mathsf{PI}\left(\right)}} \]

    12. times-frac N/A

        \[\leadsto \frac{\frac{\frac{3}{4}}{{x}^{4}} \cdot \frac{e^{{\left(\left|x\right|\right)}^{2}}}{\left|x\right|} + \left(\frac{e^{{\left(\left|x\right|\right)}^{2}}}{\left|x\right|} + \color{blue}{\frac{\frac{1}{2}}{{x}^{2}} \cdot \frac{e^{{x}^{2}}}{\left|x\right|}}\right)}{\sqrt{\mathsf{PI}\left(\right)}} \]

11. Applied rewrites 99.0%

    \[\leadsto \frac{\color{blue}{\frac{e^{x \cdot x}}{\left|x\right|} \cdot \left(\frac{0.75}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)} + \left(\frac{0.5}{x \cdot x} + 1\right)\right)}}{\sqrt{\pi}} \]

12. Final simplification 99.0%

    \[\leadsto \frac{\left(\frac{0.75}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)} + \left(1 + \frac{0.5}{x \cdot x}\right)\right) \cdot \frac{e^{x \cdot x}}{\left|x\right|}}{\sqrt{\pi}} \]
13. Add Preprocessing

Alternative 8: 99.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\left(1 - \frac{0.5}{x \cdot x}\right) \cdot \left|x\right|} \cdot \frac{1}{\frac{\sqrt{\pi}}{e^{x \cdot x}}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (/ 1.0 (* (- 1.0 (/ 0.5 (* x x))) (fabs x)))
  (/ 1.0 (/ (sqrt PI) (exp (* x x))))))

C

double code(double x) {
	return (1.0 / ((1.0 - (0.5 / (x * x))) * fabs(x))) * (1.0 / (sqrt(((double) M_PI)) / exp((x * x))));
}

Java

public static double code(double x) {
	return (1.0 / ((1.0 - (0.5 / (x * x))) * Math.abs(x))) * (1.0 / (Math.sqrt(Math.PI) / Math.exp((x * x))));
}

Python

def code(x):
	return (1.0 / ((1.0 - (0.5 / (x * x))) * math.fabs(x))) * (1.0 / (math.sqrt(math.pi) / math.exp((x * x))))

Julia

function code(x)
	return Float64(Float64(1.0 / Float64(Float64(1.0 - Float64(0.5 / Float64(x * x))) * abs(x))) * Float64(1.0 / Float64(sqrt(pi) / exp(Float64(x * x)))))
end

MATLAB

function tmp = code(x)
	tmp = (1.0 / ((1.0 - (0.5 / (x * x))) * abs(x))) * (1.0 / (sqrt(pi) / exp((x * x))));
end

Wolfram

code[x_] := N[(N[(1.0 / N[(N[(1.0 - N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(N[Sqrt[Pi], $MachinePrecision] / N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]
2. Add Preprocessing

4. Applied rewrites 33.6%

   \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{\frac{1}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)}}{\left(x \cdot x\right) \cdot \left|x\right|}, 0.125, \frac{1}{\left(x \cdot x\right) \cdot \left|x\right|}\right), \frac{1}{\mathsf{fma}\left(\frac{0.5}{\left(x \cdot x\right) \cdot \left|x\right|}, \frac{\frac{0.5}{x \cdot x} - 1}{\left|x\right|}, \frac{1}{x \cdot x}\right)}, \frac{1.875}{\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left|x\right|\right) \cdot \left(x \cdot x\right)} + \frac{0.75}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left|x\right|}\right)} \]

5. Taylor expanded in x around inf

   \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \color{blue}{\frac{1}{\left|x\right| \cdot \left(1 - \frac{1}{2} \cdot \frac{1}{{\left(\left|x\right|\right)}^{2}}\right)}} \]
6. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \color{blue}{\frac{1}{\left|x\right| \cdot \left(1 - \frac{1}{2} \cdot \frac{1}{{\left(\left|x\right|\right)}^{2}}\right)}} \]

   2. *-commutative N/A

      \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \frac{1}{\color{blue}{\left(1 - \frac{1}{2} \cdot \frac{1}{{\left(\left|x\right|\right)}^{2}}\right) \cdot \left|x\right|}} \]

   3. lower-*.f64 N/A

      \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \frac{1}{\color{blue}{\left(1 - \frac{1}{2} \cdot \frac{1}{{\left(\left|x\right|\right)}^{2}}\right) \cdot \left|x\right|}} \]

   4. lower--.f64 N/A

      \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \frac{1}{\color{blue}{\left(1 - \frac{1}{2} \cdot \frac{1}{{\left(\left|x\right|\right)}^{2}}\right)} \cdot \left|x\right|} \]

   5. associate-*r/ N/A

      \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \frac{1}{\left(1 - \color{blue}{\frac{\frac{1}{2} \cdot 1}{{\left(\left|x\right|\right)}^{2}}}\right) \cdot \left|x\right|} \]

   6. metadata-eval N/A

      \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \frac{1}{\left(1 - \frac{\color{blue}{\frac{1}{2}}}{{\left(\left|x\right|\right)}^{2}}\right) \cdot \left|x\right|} \]

   7. unpow2 N/A

      \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \frac{1}{\left(1 - \frac{\frac{1}{2}}{\color{blue}{\left|x\right| \cdot \left|x\right|}}\right) \cdot \left|x\right|} \]

   8. sqr-abs N/A

      \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \frac{1}{\left(1 - \frac{\frac{1}{2}}{\color{blue}{x \cdot x}}\right) \cdot \left|x\right|} \]

   9. unpow2 N/A

      \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \frac{1}{\left(1 - \frac{\frac{1}{2}}{\color{blue}{{x}^{2}}}\right) \cdot \left|x\right|} \]

   10. lower-/.f64 N/A

       \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \frac{1}{\left(1 - \color{blue}{\frac{\frac{1}{2}}{{x}^{2}}}\right) \cdot \left|x\right|} \]

   11. unpow2 N/A

       \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \frac{1}{\left(1 - \frac{\frac{1}{2}}{\color{blue}{x \cdot x}}\right) \cdot \left|x\right|} \]

   12. lower-*.f64 N/A

       \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \frac{1}{\left(1 - \frac{\frac{1}{2}}{\color{blue}{x \cdot x}}\right) \cdot \left|x\right|} \]

   13. lower-fabs.f64 98.9

       \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \frac{1}{\left(1 - \frac{0.5}{x \cdot x}\right) \cdot \color{blue}{\left|x\right|}} \]

7. Applied rewrites 98.9%

   \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \color{blue}{\frac{1}{\left(1 - \frac{0.5}{x \cdot x}\right) \cdot \left|x\right|}} \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right)} \cdot \frac{1}{\left(1 - \frac{\frac{1}{2}}{x \cdot x}\right) \cdot \left|x\right|} \]

   2. lift-/.f64 N/A

      \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{\mathsf{PI}\left(\right)}}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \frac{1}{\left(1 - \frac{\frac{1}{2}}{x \cdot x}\right) \cdot \left|x\right|} \]

   3. lift-*.f64 N/A

      \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\color{blue}{\left|x\right| \cdot \left|x\right|}}\right) \cdot \frac{1}{\left(1 - \frac{\frac{1}{2}}{x \cdot x}\right) \cdot \left|x\right|} \]

   4. lift-fabs.f64 N/A

      \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\color{blue}{\left|x\right|} \cdot \left|x\right|}\right) \cdot \frac{1}{\left(1 - \frac{\frac{1}{2}}{x \cdot x}\right) \cdot \left|x\right|} \]

   5. lift-fabs.f64 N/A

      \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \color{blue}{\left|x\right|}}\right) \cdot \frac{1}{\left(1 - \frac{\frac{1}{2}}{x \cdot x}\right) \cdot \left|x\right|} \]

   6. sqr-abs N/A

      \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\color{blue}{x \cdot x}}\right) \cdot \frac{1}{\left(1 - \frac{\frac{1}{2}}{x \cdot x}\right) \cdot \left|x\right|} \]

   7. lift-*.f64 N/A

      \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\color{blue}{x \cdot x}}\right) \cdot \frac{1}{\left(1 - \frac{\frac{1}{2}}{x \cdot x}\right) \cdot \left|x\right|} \]

   8. associate-/r/ N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{\mathsf{PI}\left(\right)}}{e^{x \cdot x}}}} \cdot \frac{1}{\left(1 - \frac{\frac{1}{2}}{x \cdot x}\right) \cdot \left|x\right|} \]

   9. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{\mathsf{PI}\left(\right)}}{e^{x \cdot x}}}} \cdot \frac{1}{\left(1 - \frac{\frac{1}{2}}{x \cdot x}\right) \cdot \left|x\right|} \]

   10. lower-/.f64 98.9

       \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{\pi}}{e^{x \cdot x}}}} \cdot \frac{1}{\left(1 - \frac{0.5}{x \cdot x}\right) \cdot \left|x\right|} \]

9. Applied rewrites 98.9%

   \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{\pi}}{e^{x \cdot x}}}} \cdot \frac{1}{\left(1 - \frac{0.5}{x \cdot x}\right) \cdot \left|x\right|} \]

10. Final simplification 98.9%

    \[\leadsto \frac{1}{\left(1 - \frac{0.5}{x \cdot x}\right) \cdot \left|x\right|} \cdot \frac{1}{\frac{\sqrt{\pi}}{e^{x \cdot x}}} \]
11. Add Preprocessing

Alternative 9: 99.6% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{\left(1 - \frac{0.5}{x \cdot x}\right) \cdot \left|x\right|} \cdot e^{x \cdot x}}{\sqrt{\pi}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (* (/ 1.0 (* (- 1.0 (/ 0.5 (* x x))) (fabs x))) (exp (* x x))) (sqrt PI)))

C

double code(double x) {
	return ((1.0 / ((1.0 - (0.5 / (x * x))) * fabs(x))) * exp((x * x))) / sqrt(((double) M_PI));
}

Java

public static double code(double x) {
	return ((1.0 / ((1.0 - (0.5 / (x * x))) * Math.abs(x))) * Math.exp((x * x))) / Math.sqrt(Math.PI);
}

Python

def code(x):
	return ((1.0 / ((1.0 - (0.5 / (x * x))) * math.fabs(x))) * math.exp((x * x))) / math.sqrt(math.pi)

Julia

function code(x)
	return Float64(Float64(Float64(1.0 / Float64(Float64(1.0 - Float64(0.5 / Float64(x * x))) * abs(x))) * exp(Float64(x * x))) / sqrt(pi))
end

MATLAB

function tmp = code(x)
	tmp = ((1.0 / ((1.0 - (0.5 / (x * x))) * abs(x))) * exp((x * x))) / sqrt(pi);
end

Wolfram

code[x_] := N[(N[(N[(1.0 / N[(N[(1.0 - N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]
2. Add Preprocessing

4. Applied rewrites 33.6%

   \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{\frac{1}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)}}{\left(x \cdot x\right) \cdot \left|x\right|}, 0.125, \frac{1}{\left(x \cdot x\right) \cdot \left|x\right|}\right), \frac{1}{\mathsf{fma}\left(\frac{0.5}{\left(x \cdot x\right) \cdot \left|x\right|}, \frac{\frac{0.5}{x \cdot x} - 1}{\left|x\right|}, \frac{1}{x \cdot x}\right)}, \frac{1.875}{\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left|x\right|\right) \cdot \left(x \cdot x\right)} + \frac{0.75}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left|x\right|}\right)} \]

5. Taylor expanded in x around inf

   \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \color{blue}{\frac{1}{\left|x\right| \cdot \left(1 - \frac{1}{2} \cdot \frac{1}{{\left(\left|x\right|\right)}^{2}}\right)}} \]
6. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \color{blue}{\frac{1}{\left|x\right| \cdot \left(1 - \frac{1}{2} \cdot \frac{1}{{\left(\left|x\right|\right)}^{2}}\right)}} \]

   2. *-commutative N/A

      \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \frac{1}{\color{blue}{\left(1 - \frac{1}{2} \cdot \frac{1}{{\left(\left|x\right|\right)}^{2}}\right) \cdot \left|x\right|}} \]

   3. lower-*.f64 N/A

      \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \frac{1}{\color{blue}{\left(1 - \frac{1}{2} \cdot \frac{1}{{\left(\left|x\right|\right)}^{2}}\right) \cdot \left|x\right|}} \]

   4. lower--.f64 N/A

      \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \frac{1}{\color{blue}{\left(1 - \frac{1}{2} \cdot \frac{1}{{\left(\left|x\right|\right)}^{2}}\right)} \cdot \left|x\right|} \]

   5. associate-*r/ N/A

      \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \frac{1}{\left(1 - \color{blue}{\frac{\frac{1}{2} \cdot 1}{{\left(\left|x\right|\right)}^{2}}}\right) \cdot \left|x\right|} \]

   6. metadata-eval N/A

      \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \frac{1}{\left(1 - \frac{\color{blue}{\frac{1}{2}}}{{\left(\left|x\right|\right)}^{2}}\right) \cdot \left|x\right|} \]

   7. unpow2 N/A

      \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \frac{1}{\left(1 - \frac{\frac{1}{2}}{\color{blue}{\left|x\right| \cdot \left|x\right|}}\right) \cdot \left|x\right|} \]

   8. sqr-abs N/A

      \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \frac{1}{\left(1 - \frac{\frac{1}{2}}{\color{blue}{x \cdot x}}\right) \cdot \left|x\right|} \]

   9. unpow2 N/A

      \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \frac{1}{\left(1 - \frac{\frac{1}{2}}{\color{blue}{{x}^{2}}}\right) \cdot \left|x\right|} \]

   10. lower-/.f64 N/A

       \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \frac{1}{\left(1 - \color{blue}{\frac{\frac{1}{2}}{{x}^{2}}}\right) \cdot \left|x\right|} \]

   11. unpow2 N/A

       \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \frac{1}{\left(1 - \frac{\frac{1}{2}}{\color{blue}{x \cdot x}}\right) \cdot \left|x\right|} \]

   12. lower-*.f64 N/A

       \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \frac{1}{\left(1 - \frac{\frac{1}{2}}{\color{blue}{x \cdot x}}\right) \cdot \left|x\right|} \]

   13. lower-fabs.f64 98.9

       \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \frac{1}{\left(1 - \frac{0.5}{x \cdot x}\right) \cdot \color{blue}{\left|x\right|}} \]

7. Applied rewrites 98.9%

   \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \color{blue}{\frac{1}{\left(1 - \frac{0.5}{x \cdot x}\right) \cdot \left|x\right|}} \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \frac{1}{\left(1 - \frac{\frac{1}{2}}{x \cdot x}\right) \cdot \left|x\right|}} \]

9. Applied rewrites 98.9%

   \[\leadsto \color{blue}{\frac{\frac{1}{\left(1 - \frac{0.5}{x \cdot x}\right) \cdot \left|x\right|} \cdot e^{x \cdot x}}{\sqrt{\pi}}} \]
10. Add Preprocessing

Alternative 10: 99.6% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{e^{x \cdot x}}{\left|x\right|} \cdot \left(1 + \frac{0.5}{x \cdot x}\right)}{\sqrt{\pi}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (* (/ (exp (* x x)) (fabs x)) (+ 1.0 (/ 0.5 (* x x)))) (sqrt PI)))

C

double code(double x) {
	return ((exp((x * x)) / fabs(x)) * (1.0 + (0.5 / (x * x)))) / sqrt(((double) M_PI));
}

Java

public static double code(double x) {
	return ((Math.exp((x * x)) / Math.abs(x)) * (1.0 + (0.5 / (x * x)))) / Math.sqrt(Math.PI);
}

Python

def code(x):
	return ((math.exp((x * x)) / math.fabs(x)) * (1.0 + (0.5 / (x * x)))) / math.sqrt(math.pi)

Julia

function code(x)
	return Float64(Float64(Float64(exp(Float64(x * x)) / abs(x)) * Float64(1.0 + Float64(0.5 / Float64(x * x)))) / sqrt(pi))
end

MATLAB

function tmp = code(x)
	tmp = ((exp((x * x)) / abs(x)) * (1.0 + (0.5 / (x * x)))) / sqrt(pi);
end

Wolfram

code[x_] := N[(N[(N[(N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision] / N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]
2. Add Preprocessing

4. Applied rewrites 100.0%

   \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \color{blue}{\left(\frac{1}{\left|x\right|} + \left(\frac{0.5}{\left(x \cdot x\right) \cdot \left|x\right|} + \left(\frac{1.875}{\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left|x\right|\right) \cdot \left(x \cdot x\right)} + \frac{0.75}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left|x\right|}\right)\right)\right)} \]

6. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\frac{e^{x \cdot x} \cdot \left(\frac{\frac{0.5}{x \cdot x} + 1}{\left|x\right|} + \left(\frac{0.75}{\left(\left|x\right| \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)} + \frac{1.875}{\left(\left(\left|x\right| \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(x \cdot x\right)}\right)\right)}{\sqrt{\pi}}} \]

7. Taylor expanded in x around inf

   \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot \frac{e^{{x}^{2}}}{{x}^{2} \cdot \left|x\right|} + \frac{e^{{x}^{2}}}{\left|x\right|}}}{\sqrt{\mathsf{PI}\left(\right)}} \]
8. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \frac{\color{blue}{\frac{\frac{1}{2} \cdot e^{{x}^{2}}}{{x}^{2} \cdot \left|x\right|}} + \frac{e^{{x}^{2}}}{\left|x\right|}}{\sqrt{\mathsf{PI}\left(\right)}} \]

   2. times-frac N/A

      \[\leadsto \frac{\color{blue}{\frac{\frac{1}{2}}{{x}^{2}} \cdot \frac{e^{{x}^{2}}}{\left|x\right|}} + \frac{e^{{x}^{2}}}{\left|x\right|}}{\sqrt{\mathsf{PI}\left(\right)}} \]

   3. metadata-eval N/A

      \[\leadsto \frac{\frac{\color{blue}{\frac{1}{2} \cdot 1}}{{x}^{2}} \cdot \frac{e^{{x}^{2}}}{\left|x\right|} + \frac{e^{{x}^{2}}}{\left|x\right|}}{\sqrt{\mathsf{PI}\left(\right)}} \]

   4. unpow2 N/A

      \[\leadsto \frac{\frac{\frac{1}{2} \cdot 1}{\color{blue}{x \cdot x}} \cdot \frac{e^{{x}^{2}}}{\left|x\right|} + \frac{e^{{x}^{2}}}{\left|x\right|}}{\sqrt{\mathsf{PI}\left(\right)}} \]

   5. sqr-abs N/A

      \[\leadsto \frac{\frac{\frac{1}{2} \cdot 1}{\color{blue}{\left|x\right| \cdot \left|x\right|}} \cdot \frac{e^{{x}^{2}}}{\left|x\right|} + \frac{e^{{x}^{2}}}{\left|x\right|}}{\sqrt{\mathsf{PI}\left(\right)}} \]

   6. unpow2 N/A

      \[\leadsto \frac{\frac{\frac{1}{2} \cdot 1}{\color{blue}{{\left(\left|x\right|\right)}^{2}}} \cdot \frac{e^{{x}^{2}}}{\left|x\right|} + \frac{e^{{x}^{2}}}{\left|x\right|}}{\sqrt{\mathsf{PI}\left(\right)}} \]

   7. associate-*r/ N/A

      \[\leadsto \frac{\color{blue}{\left(\frac{1}{2} \cdot \frac{1}{{\left(\left|x\right|\right)}^{2}}\right)} \cdot \frac{e^{{x}^{2}}}{\left|x\right|} + \frac{e^{{x}^{2}}}{\left|x\right|}}{\sqrt{\mathsf{PI}\left(\right)}} \]

   8. unpow2 N/A

      \[\leadsto \frac{\left(\frac{1}{2} \cdot \frac{1}{{\left(\left|x\right|\right)}^{2}}\right) \cdot \frac{e^{\color{blue}{x \cdot x}}}{\left|x\right|} + \frac{e^{{x}^{2}}}{\left|x\right|}}{\sqrt{\mathsf{PI}\left(\right)}} \]

   9. sqr-abs N/A

      \[\leadsto \frac{\left(\frac{1}{2} \cdot \frac{1}{{\left(\left|x\right|\right)}^{2}}\right) \cdot \frac{e^{\color{blue}{\left|x\right| \cdot \left|x\right|}}}{\left|x\right|} + \frac{e^{{x}^{2}}}{\left|x\right|}}{\sqrt{\mathsf{PI}\left(\right)}} \]

   10. unpow2 N/A

       \[\leadsto \frac{\left(\frac{1}{2} \cdot \frac{1}{{\left(\left|x\right|\right)}^{2}}\right) \cdot \frac{e^{\color{blue}{{\left(\left|x\right|\right)}^{2}}}}{\left|x\right|} + \frac{e^{{x}^{2}}}{\left|x\right|}}{\sqrt{\mathsf{PI}\left(\right)}} \]

   11. unpow2 N/A

       \[\leadsto \frac{\left(\frac{1}{2} \cdot \frac{1}{{\left(\left|x\right|\right)}^{2}}\right) \cdot \frac{e^{{\left(\left|x\right|\right)}^{2}}}{\left|x\right|} + \frac{e^{\color{blue}{x \cdot x}}}{\left|x\right|}}{\sqrt{\mathsf{PI}\left(\right)}} \]

   12. sqr-abs N/A

       \[\leadsto \frac{\left(\frac{1}{2} \cdot \frac{1}{{\left(\left|x\right|\right)}^{2}}\right) \cdot \frac{e^{{\left(\left|x\right|\right)}^{2}}}{\left|x\right|} + \frac{e^{\color{blue}{\left|x\right| \cdot \left|x\right|}}}{\left|x\right|}}{\sqrt{\mathsf{PI}\left(\right)}} \]

   13. unpow2 N/A

       \[\leadsto \frac{\left(\frac{1}{2} \cdot \frac{1}{{\left(\left|x\right|\right)}^{2}}\right) \cdot \frac{e^{{\left(\left|x\right|\right)}^{2}}}{\left|x\right|} + \frac{e^{\color{blue}{{\left(\left|x\right|\right)}^{2}}}}{\left|x\right|}}{\sqrt{\mathsf{PI}\left(\right)}} \]

   14. distribute-lft1-in N/A

       \[\leadsto \frac{\color{blue}{\left(\frac{1}{2} \cdot \frac{1}{{\left(\left|x\right|\right)}^{2}} + 1\right) \cdot \frac{e^{{\left(\left|x\right|\right)}^{2}}}{\left|x\right|}}}{\sqrt{\mathsf{PI}\left(\right)}} \]

   15. lower-*.f64 N/A

       \[\leadsto \frac{\color{blue}{\left(\frac{1}{2} \cdot \frac{1}{{\left(\left|x\right|\right)}^{2}} + 1\right) \cdot \frac{e^{{\left(\left|x\right|\right)}^{2}}}{\left|x\right|}}}{\sqrt{\mathsf{PI}\left(\right)}} \]

9. Applied rewrites 98.9%

   \[\leadsto \frac{\color{blue}{\left(\frac{0.5}{x \cdot x} + 1\right) \cdot \frac{e^{x \cdot x}}{\left|x\right|}}}{\sqrt{\pi}} \]

10. Final simplification 98.9%

    \[\leadsto \frac{\frac{e^{x \cdot x}}{\left|x\right|} \cdot \left(1 + \frac{0.5}{x \cdot x}\right)}{\sqrt{\pi}} \]
11. Add Preprocessing

Alternative 11: 99.6% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{e^{x \cdot x}}{\left|x\right|}}{\sqrt{\pi}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (/ (exp (* x x)) (fabs x)) (sqrt PI)))

C

double code(double x) {
	return (exp((x * x)) / fabs(x)) / sqrt(((double) M_PI));
}

Java

public static double code(double x) {
	return (Math.exp((x * x)) / Math.abs(x)) / Math.sqrt(Math.PI);
}

Python

def code(x):
	return (math.exp((x * x)) / math.fabs(x)) / math.sqrt(math.pi)

Julia

function code(x)
	return Float64(Float64(exp(Float64(x * x)) / abs(x)) / sqrt(pi))
end

MATLAB

function tmp = code(x)
	tmp = (exp((x * x)) / abs(x)) / sqrt(pi);
end

Wolfram

code[x_] := N[(N[(N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision] / N[Abs[x], $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]
2. Add Preprocessing

4. Applied rewrites 100.0%

   \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \color{blue}{\left(\frac{1}{\left|x\right|} + \left(\frac{0.5}{\left(x \cdot x\right) \cdot \left|x\right|} + \left(\frac{1.875}{\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left|x\right|\right) \cdot \left(x \cdot x\right)} + \frac{0.75}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left|x\right|}\right)\right)\right)} \]

6. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\frac{e^{x \cdot x} \cdot \left(\frac{\frac{0.5}{x \cdot x} + 1}{\left|x\right|} + \left(\frac{0.75}{\left(\left|x\right| \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)} + \frac{1.875}{\left(\left(\left|x\right| \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(x \cdot x\right)}\right)\right)}{\sqrt{\pi}}} \]

7. Taylor expanded in x around inf

   \[\leadsto \frac{\color{blue}{\frac{e^{{x}^{2}}}{\left|x\right|}}}{\sqrt{\mathsf{PI}\left(\right)}} \]
8. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \frac{\frac{e^{\color{blue}{x \cdot x}}}{\left|x\right|}}{\sqrt{\mathsf{PI}\left(\right)}} \]

   2. sqr-abs N/A

      \[\leadsto \frac{\frac{e^{\color{blue}{\left|x\right| \cdot \left|x\right|}}}{\left|x\right|}}{\sqrt{\mathsf{PI}\left(\right)}} \]

   3. unpow2 N/A

      \[\leadsto \frac{\frac{e^{\color{blue}{{\left(\left|x\right|\right)}^{2}}}}{\left|x\right|}}{\sqrt{\mathsf{PI}\left(\right)}} \]

   4. lower-/.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{e^{{\left(\left|x\right|\right)}^{2}}}{\left|x\right|}}}{\sqrt{\mathsf{PI}\left(\right)}} \]

   5. unpow2 N/A

      \[\leadsto \frac{\frac{e^{\color{blue}{\left|x\right| \cdot \left|x\right|}}}{\left|x\right|}}{\sqrt{\mathsf{PI}\left(\right)}} \]

   6. sqr-abs N/A

      \[\leadsto \frac{\frac{e^{\color{blue}{x \cdot x}}}{\left|x\right|}}{\sqrt{\mathsf{PI}\left(\right)}} \]

   7. unpow2 N/A

      \[\leadsto \frac{\frac{e^{\color{blue}{{x}^{2}}}}{\left|x\right|}}{\sqrt{\mathsf{PI}\left(\right)}} \]

   8. lower-exp.f64 N/A

      \[\leadsto \frac{\frac{\color{blue}{e^{{x}^{2}}}}{\left|x\right|}}{\sqrt{\mathsf{PI}\left(\right)}} \]

   9. unpow2 N/A

      \[\leadsto \frac{\frac{e^{\color{blue}{x \cdot x}}}{\left|x\right|}}{\sqrt{\mathsf{PI}\left(\right)}} \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{\frac{e^{\color{blue}{x \cdot x}}}{\left|x\right|}}{\sqrt{\mathsf{PI}\left(\right)}} \]

   11. lower-fabs.f64 98.8

       \[\leadsto \frac{\frac{e^{x \cdot x}}{\color{blue}{\left|x\right|}}}{\sqrt{\pi}} \]

9. Applied rewrites 98.8%

   \[\leadsto \frac{\color{blue}{\frac{e^{x \cdot x}}{\left|x\right|}}}{\sqrt{\pi}} \]
10. Add Preprocessing

Alternative 12: 99.6% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \frac{e^{x \cdot x}}{\sqrt{\pi} \cdot \left|x\right|} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (exp (* x x)) (* (sqrt PI) (fabs x))))

C

double code(double x) {
	return exp((x * x)) / (sqrt(((double) M_PI)) * fabs(x));
}

Java

public static double code(double x) {
	return Math.exp((x * x)) / (Math.sqrt(Math.PI) * Math.abs(x));
}

Python

def code(x):
	return math.exp((x * x)) / (math.sqrt(math.pi) * math.fabs(x))

Julia

function code(x)
	return Float64(exp(Float64(x * x)) / Float64(sqrt(pi) * abs(x)))
end

MATLAB

function tmp = code(x)
	tmp = exp((x * x)) / (sqrt(pi) * abs(x));
end

Wolfram

code[x_] := N[(N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision] / N[(N[Sqrt[Pi], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]
2. Add Preprocessing

4. Applied rewrites 100.0%

   \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \color{blue}{\left(\frac{\frac{0.5}{x \cdot x} + 1}{\left|x\right|} + \left(\frac{1.875}{\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left|x\right|\right) \cdot \left(x \cdot x\right)} + \frac{0.75}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left|x\right|}\right)\right)} \]

5. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \frac{e^{{\left(\left|x\right|\right)}^{2}}}{\left|x\right|}} \]
6. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot e^{{\left(\left|x\right|\right)}^{2}}}{\left|x\right|}} \]

   2. *-commutative N/A

      \[\leadsto \frac{\color{blue}{e^{{\left(\left|x\right|\right)}^{2}} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}}}{\left|x\right|} \]

   3. associate-/l* N/A

      \[\leadsto \color{blue}{e^{{\left(\left|x\right|\right)}^{2}} \cdot \frac{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}}}{\left|x\right|}} \]

   4. *-commutative N/A

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}}}{\left|x\right|} \cdot e^{{\left(\left|x\right|\right)}^{2}}} \]

   5. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}}}{\left|x\right|} \cdot e^{{\left(\left|x\right|\right)}^{2}}} \]

   6. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}}}{\left|x\right|}} \cdot e^{{\left(\left|x\right|\right)}^{2}} \]

   7. lower-sqrt.f64 N/A

      \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}}}}{\left|x\right|} \cdot e^{{\left(\left|x\right|\right)}^{2}} \]

   8. lower-/.f64 N/A

      \[\leadsto \frac{\sqrt{\color{blue}{\frac{1}{\mathsf{PI}\left(\right)}}}}{\left|x\right|} \cdot e^{{\left(\left|x\right|\right)}^{2}} \]

   9. lower-PI.f64 N/A

      \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{\mathsf{PI}\left(\right)}}}}{\left|x\right|} \cdot e^{{\left(\left|x\right|\right)}^{2}} \]

   10. lower-fabs.f64 N/A

       \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}}}{\color{blue}{\left|x\right|}} \cdot e^{{\left(\left|x\right|\right)}^{2}} \]

   11. unpow2 N/A

       \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}}}{\left|x\right|} \cdot e^{\color{blue}{\left|x\right| \cdot \left|x\right|}} \]

   12. sqr-abs N/A

       \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}}}{\left|x\right|} \cdot e^{\color{blue}{x \cdot x}} \]

   13. unpow2 N/A

       \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}}}{\left|x\right|} \cdot e^{\color{blue}{{x}^{2}}} \]

   14. lower-exp.f64 N/A

       \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}}}{\left|x\right|} \cdot \color{blue}{e^{{x}^{2}}} \]

   15. unpow2 N/A

       \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}}}{\left|x\right|} \cdot e^{\color{blue}{x \cdot x}} \]

   16. lower-*.f64 98.8

       \[\leadsto \frac{\sqrt{\frac{1}{\pi}}}{\left|x\right|} \cdot e^{\color{blue}{x \cdot x}} \]

7. Applied rewrites 98.8%

   \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\pi}}}{\left|x\right|} \cdot e^{x \cdot x}} \]
8. Step-by-step derivation

9. Applied rewrites 98.8%

   \[\leadsto \color{blue}{\frac{e^{x \cdot x}}{\sqrt{\pi} \cdot \left|x\right|}} \]
10. Add Preprocessing

Alternative 13: 83.6% accurate, 5.1× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x \cdot x, 0.5\right), x \cdot x, 1\right), x \cdot x, 1\right)}{\left|x\right|} \cdot \left(1 + \frac{0.5}{x \cdot x}\right)}{\sqrt{\pi}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  (*
   (/
    (fma (fma (fma 0.16666666666666666 (* x x) 0.5) (* x x) 1.0) (* x x) 1.0)
    (fabs x))
   (+ 1.0 (/ 0.5 (* x x))))
  (sqrt PI)))

C

double code(double x) {
	return ((fma(fma(fma(0.16666666666666666, (x * x), 0.5), (x * x), 1.0), (x * x), 1.0) / fabs(x)) * (1.0 + (0.5 / (x * x)))) / sqrt(((double) M_PI));
}

Julia

function code(x)
	return Float64(Float64(Float64(fma(fma(fma(0.16666666666666666, Float64(x * x), 0.5), Float64(x * x), 1.0), Float64(x * x), 1.0) / abs(x)) * Float64(1.0 + Float64(0.5 / Float64(x * x)))) / sqrt(pi))
end

Wolfram

code[x_] := N[(N[(N[(N[(N[(N[(0.16666666666666666 * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] / N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]
2. Add Preprocessing

4. Applied rewrites 100.0%

   \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \color{blue}{\left(\frac{1}{\left|x\right|} + \left(\frac{0.5}{\left(x \cdot x\right) \cdot \left|x\right|} + \left(\frac{1.875}{\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left|x\right|\right) \cdot \left(x \cdot x\right)} + \frac{0.75}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left|x\right|}\right)\right)\right)} \]

6. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\frac{e^{x \cdot x} \cdot \left(\frac{\frac{0.5}{x \cdot x} + 1}{\left|x\right|} + \left(\frac{0.75}{\left(\left|x\right| \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)} + \frac{1.875}{\left(\left(\left|x\right| \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(x \cdot x\right)}\right)\right)}{\sqrt{\pi}}} \]

7. Taylor expanded in x around inf

   \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot \frac{e^{{x}^{2}}}{{x}^{2} \cdot \left|x\right|} + \frac{e^{{x}^{2}}}{\left|x\right|}}}{\sqrt{\mathsf{PI}\left(\right)}} \]
8. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \frac{\color{blue}{\frac{\frac{1}{2} \cdot e^{{x}^{2}}}{{x}^{2} \cdot \left|x\right|}} + \frac{e^{{x}^{2}}}{\left|x\right|}}{\sqrt{\mathsf{PI}\left(\right)}} \]

   2. times-frac N/A

      \[\leadsto \frac{\color{blue}{\frac{\frac{1}{2}}{{x}^{2}} \cdot \frac{e^{{x}^{2}}}{\left|x\right|}} + \frac{e^{{x}^{2}}}{\left|x\right|}}{\sqrt{\mathsf{PI}\left(\right)}} \]

   3. metadata-eval N/A

      \[\leadsto \frac{\frac{\color{blue}{\frac{1}{2} \cdot 1}}{{x}^{2}} \cdot \frac{e^{{x}^{2}}}{\left|x\right|} + \frac{e^{{x}^{2}}}{\left|x\right|}}{\sqrt{\mathsf{PI}\left(\right)}} \]

   4. unpow2 N/A

      \[\leadsto \frac{\frac{\frac{1}{2} \cdot 1}{\color{blue}{x \cdot x}} \cdot \frac{e^{{x}^{2}}}{\left|x\right|} + \frac{e^{{x}^{2}}}{\left|x\right|}}{\sqrt{\mathsf{PI}\left(\right)}} \]

   5. sqr-abs N/A

      \[\leadsto \frac{\frac{\frac{1}{2} \cdot 1}{\color{blue}{\left|x\right| \cdot \left|x\right|}} \cdot \frac{e^{{x}^{2}}}{\left|x\right|} + \frac{e^{{x}^{2}}}{\left|x\right|}}{\sqrt{\mathsf{PI}\left(\right)}} \]

   6. unpow2 N/A

      \[\leadsto \frac{\frac{\frac{1}{2} \cdot 1}{\color{blue}{{\left(\left|x\right|\right)}^{2}}} \cdot \frac{e^{{x}^{2}}}{\left|x\right|} + \frac{e^{{x}^{2}}}{\left|x\right|}}{\sqrt{\mathsf{PI}\left(\right)}} \]

   7. associate-*r/ N/A

      \[\leadsto \frac{\color{blue}{\left(\frac{1}{2} \cdot \frac{1}{{\left(\left|x\right|\right)}^{2}}\right)} \cdot \frac{e^{{x}^{2}}}{\left|x\right|} + \frac{e^{{x}^{2}}}{\left|x\right|}}{\sqrt{\mathsf{PI}\left(\right)}} \]

   8. unpow2 N/A

      \[\leadsto \frac{\left(\frac{1}{2} \cdot \frac{1}{{\left(\left|x\right|\right)}^{2}}\right) \cdot \frac{e^{\color{blue}{x \cdot x}}}{\left|x\right|} + \frac{e^{{x}^{2}}}{\left|x\right|}}{\sqrt{\mathsf{PI}\left(\right)}} \]

   9. sqr-abs N/A

      \[\leadsto \frac{\left(\frac{1}{2} \cdot \frac{1}{{\left(\left|x\right|\right)}^{2}}\right) \cdot \frac{e^{\color{blue}{\left|x\right| \cdot \left|x\right|}}}{\left|x\right|} + \frac{e^{{x}^{2}}}{\left|x\right|}}{\sqrt{\mathsf{PI}\left(\right)}} \]

   10. unpow2 N/A

       \[\leadsto \frac{\left(\frac{1}{2} \cdot \frac{1}{{\left(\left|x\right|\right)}^{2}}\right) \cdot \frac{e^{\color{blue}{{\left(\left|x\right|\right)}^{2}}}}{\left|x\right|} + \frac{e^{{x}^{2}}}{\left|x\right|}}{\sqrt{\mathsf{PI}\left(\right)}} \]

   11. unpow2 N/A

       \[\leadsto \frac{\left(\frac{1}{2} \cdot \frac{1}{{\left(\left|x\right|\right)}^{2}}\right) \cdot \frac{e^{{\left(\left|x\right|\right)}^{2}}}{\left|x\right|} + \frac{e^{\color{blue}{x \cdot x}}}{\left|x\right|}}{\sqrt{\mathsf{PI}\left(\right)}} \]

   12. sqr-abs N/A

       \[\leadsto \frac{\left(\frac{1}{2} \cdot \frac{1}{{\left(\left|x\right|\right)}^{2}}\right) \cdot \frac{e^{{\left(\left|x\right|\right)}^{2}}}{\left|x\right|} + \frac{e^{\color{blue}{\left|x\right| \cdot \left|x\right|}}}{\left|x\right|}}{\sqrt{\mathsf{PI}\left(\right)}} \]

   13. unpow2 N/A

       \[\leadsto \frac{\left(\frac{1}{2} \cdot \frac{1}{{\left(\left|x\right|\right)}^{2}}\right) \cdot \frac{e^{{\left(\left|x\right|\right)}^{2}}}{\left|x\right|} + \frac{e^{\color{blue}{{\left(\left|x\right|\right)}^{2}}}}{\left|x\right|}}{\sqrt{\mathsf{PI}\left(\right)}} \]

   14. distribute-lft1-in N/A

       \[\leadsto \frac{\color{blue}{\left(\frac{1}{2} \cdot \frac{1}{{\left(\left|x\right|\right)}^{2}} + 1\right) \cdot \frac{e^{{\left(\left|x\right|\right)}^{2}}}{\left|x\right|}}}{\sqrt{\mathsf{PI}\left(\right)}} \]

   15. lower-*.f64 N/A

       \[\leadsto \frac{\color{blue}{\left(\frac{1}{2} \cdot \frac{1}{{\left(\left|x\right|\right)}^{2}} + 1\right) \cdot \frac{e^{{\left(\left|x\right|\right)}^{2}}}{\left|x\right|}}}{\sqrt{\mathsf{PI}\left(\right)}} \]

9. Applied rewrites 98.9%

   \[\leadsto \frac{\color{blue}{\left(\frac{0.5}{x \cdot x} + 1\right) \cdot \frac{e^{x \cdot x}}{\left|x\right|}}}{\sqrt{\pi}} \]

10. Taylor expanded in x around 0

    \[\leadsto \frac{\left(\frac{\frac{1}{2}}{x \cdot x} + 1\right) \cdot \frac{1 + {x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)\right)}{\left|\color{blue}{x}\right|}}{\sqrt{\mathsf{PI}\left(\right)}} \]
11. Step-by-step derivation

12. Applied rewrites 84.8%

    \[\leadsto \frac{\left(\frac{0.5}{x \cdot x} + 1\right) \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x \cdot x, 0.5\right), x \cdot x, 1\right), x \cdot x, 1\right)}{\left|\color{blue}{x}\right|}}{\sqrt{\pi}} \]

13. Final simplification 84.8%

    \[\leadsto \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x \cdot x, 0.5\right), x \cdot x, 1\right), x \cdot x, 1\right)}{\left|x\right|} \cdot \left(1 + \frac{0.5}{x \cdot x}\right)}{\sqrt{\pi}} \]
14. Add Preprocessing

Alternative 14: 83.6% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x \cdot x, 0.5\right), x \cdot x, 1\right), x \cdot x, 1\right)}{\sqrt{\pi} \cdot \left|x\right|} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  (fma (fma (fma 0.16666666666666666 (* x x) 0.5) (* x x) 1.0) (* x x) 1.0)
  (* (sqrt PI) (fabs x))))

C

double code(double x) {
	return fma(fma(fma(0.16666666666666666, (x * x), 0.5), (x * x), 1.0), (x * x), 1.0) / (sqrt(((double) M_PI)) * fabs(x));
}

Julia

function code(x)
	return Float64(fma(fma(fma(0.16666666666666666, Float64(x * x), 0.5), Float64(x * x), 1.0), Float64(x * x), 1.0) / Float64(sqrt(pi) * abs(x)))
end

Wolfram

code[x_] := N[(N[(N[(N[(0.16666666666666666 * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] / N[(N[Sqrt[Pi], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]
2. Add Preprocessing

4. Applied rewrites 100.0%

   \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \color{blue}{\left(\frac{\frac{0.5}{x \cdot x} + 1}{\left|x\right|} + \left(\frac{1.875}{\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left|x\right|\right) \cdot \left(x \cdot x\right)} + \frac{0.75}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left|x\right|}\right)\right)} \]

5. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \frac{e^{{\left(\left|x\right|\right)}^{2}}}{\left|x\right|}} \]
6. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot e^{{\left(\left|x\right|\right)}^{2}}}{\left|x\right|}} \]

   2. *-commutative N/A

      \[\leadsto \frac{\color{blue}{e^{{\left(\left|x\right|\right)}^{2}} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}}}{\left|x\right|} \]

   3. associate-/l* N/A

      \[\leadsto \color{blue}{e^{{\left(\left|x\right|\right)}^{2}} \cdot \frac{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}}}{\left|x\right|}} \]

   4. *-commutative N/A

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}}}{\left|x\right|} \cdot e^{{\left(\left|x\right|\right)}^{2}}} \]

   5. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}}}{\left|x\right|} \cdot e^{{\left(\left|x\right|\right)}^{2}}} \]

   6. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}}}{\left|x\right|}} \cdot e^{{\left(\left|x\right|\right)}^{2}} \]

   7. lower-sqrt.f64 N/A

      \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}}}}{\left|x\right|} \cdot e^{{\left(\left|x\right|\right)}^{2}} \]

   8. lower-/.f64 N/A

      \[\leadsto \frac{\sqrt{\color{blue}{\frac{1}{\mathsf{PI}\left(\right)}}}}{\left|x\right|} \cdot e^{{\left(\left|x\right|\right)}^{2}} \]

   9. lower-PI.f64 N/A

      \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{\mathsf{PI}\left(\right)}}}}{\left|x\right|} \cdot e^{{\left(\left|x\right|\right)}^{2}} \]

   10. lower-fabs.f64 N/A

       \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}}}{\color{blue}{\left|x\right|}} \cdot e^{{\left(\left|x\right|\right)}^{2}} \]

   11. unpow2 N/A

       \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}}}{\left|x\right|} \cdot e^{\color{blue}{\left|x\right| \cdot \left|x\right|}} \]

   12. sqr-abs N/A

       \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}}}{\left|x\right|} \cdot e^{\color{blue}{x \cdot x}} \]

   13. unpow2 N/A

       \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}}}{\left|x\right|} \cdot e^{\color{blue}{{x}^{2}}} \]

   14. lower-exp.f64 N/A

       \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}}}{\left|x\right|} \cdot \color{blue}{e^{{x}^{2}}} \]

   15. unpow2 N/A

       \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}}}{\left|x\right|} \cdot e^{\color{blue}{x \cdot x}} \]

   16. lower-*.f64 98.8

       \[\leadsto \frac{\sqrt{\frac{1}{\pi}}}{\left|x\right|} \cdot e^{\color{blue}{x \cdot x}} \]

7. Applied rewrites 98.8%

   \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\pi}}}{\left|x\right|} \cdot e^{x \cdot x}} \]
8. Step-by-step derivation

9. Applied rewrites 98.8%

   \[\leadsto \color{blue}{\frac{e^{x \cdot x}}{\sqrt{\pi} \cdot \left|x\right|}} \]

10. Taylor expanded in x around 0

    \[\leadsto \frac{1 + {x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)\right)}{\color{blue}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left|x\right|} \]
11. Step-by-step derivation

12. Applied rewrites 84.8%

    \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x \cdot x, 0.5\right), x \cdot x, 1\right), x \cdot x, 1\right)}{\color{blue}{\sqrt{\pi}} \cdot \left|x\right|} \]
13. Add Preprocessing

Alternative 15: 75.6% accurate, 9.1× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.5, x \cdot x, 1\right), x \cdot x, 1\right)}{\sqrt{\pi} \cdot \left|x\right|} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (fma (fma 0.5 (* x x) 1.0) (* x x) 1.0) (* (sqrt PI) (fabs x))))

C

double code(double x) {
	return fma(fma(0.5, (x * x), 1.0), (x * x), 1.0) / (sqrt(((double) M_PI)) * fabs(x));
}

Julia

function code(x)
	return Float64(fma(fma(0.5, Float64(x * x), 1.0), Float64(x * x), 1.0) / Float64(sqrt(pi) * abs(x)))
end

Wolfram

code[x_] := N[(N[(N[(0.5 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] / N[(N[Sqrt[Pi], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]
2. Add Preprocessing

4. Applied rewrites 100.0%

   \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \color{blue}{\left(\frac{\frac{0.5}{x \cdot x} + 1}{\left|x\right|} + \left(\frac{1.875}{\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left|x\right|\right) \cdot \left(x \cdot x\right)} + \frac{0.75}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left|x\right|}\right)\right)} \]

5. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \frac{e^{{\left(\left|x\right|\right)}^{2}}}{\left|x\right|}} \]
6. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot e^{{\left(\left|x\right|\right)}^{2}}}{\left|x\right|}} \]

   2. *-commutative N/A

      \[\leadsto \frac{\color{blue}{e^{{\left(\left|x\right|\right)}^{2}} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}}}{\left|x\right|} \]

   3. associate-/l* N/A

      \[\leadsto \color{blue}{e^{{\left(\left|x\right|\right)}^{2}} \cdot \frac{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}}}{\left|x\right|}} \]

   4. *-commutative N/A

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}}}{\left|x\right|} \cdot e^{{\left(\left|x\right|\right)}^{2}}} \]

   5. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}}}{\left|x\right|} \cdot e^{{\left(\left|x\right|\right)}^{2}}} \]

   6. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}}}{\left|x\right|}} \cdot e^{{\left(\left|x\right|\right)}^{2}} \]

   7. lower-sqrt.f64 N/A

      \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}}}}{\left|x\right|} \cdot e^{{\left(\left|x\right|\right)}^{2}} \]

   8. lower-/.f64 N/A

      \[\leadsto \frac{\sqrt{\color{blue}{\frac{1}{\mathsf{PI}\left(\right)}}}}{\left|x\right|} \cdot e^{{\left(\left|x\right|\right)}^{2}} \]

   9. lower-PI.f64 N/A

      \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{\mathsf{PI}\left(\right)}}}}{\left|x\right|} \cdot e^{{\left(\left|x\right|\right)}^{2}} \]

   10. lower-fabs.f64 N/A

       \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}}}{\color{blue}{\left|x\right|}} \cdot e^{{\left(\left|x\right|\right)}^{2}} \]

   11. unpow2 N/A

       \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}}}{\left|x\right|} \cdot e^{\color{blue}{\left|x\right| \cdot \left|x\right|}} \]

   12. sqr-abs N/A

       \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}}}{\left|x\right|} \cdot e^{\color{blue}{x \cdot x}} \]

   13. unpow2 N/A

       \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}}}{\left|x\right|} \cdot e^{\color{blue}{{x}^{2}}} \]

   14. lower-exp.f64 N/A

       \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}}}{\left|x\right|} \cdot \color{blue}{e^{{x}^{2}}} \]

   15. unpow2 N/A

       \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}}}{\left|x\right|} \cdot e^{\color{blue}{x \cdot x}} \]

   16. lower-*.f64 98.8

       \[\leadsto \frac{\sqrt{\frac{1}{\pi}}}{\left|x\right|} \cdot e^{\color{blue}{x \cdot x}} \]

7. Applied rewrites 98.8%

   \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\pi}}}{\left|x\right|} \cdot e^{x \cdot x}} \]
8. Step-by-step derivation

9. Applied rewrites 98.8%

   \[\leadsto \color{blue}{\frac{e^{x \cdot x}}{\sqrt{\pi} \cdot \left|x\right|}} \]

10. Taylor expanded in x around 0

    \[\leadsto \frac{1 + {x}^{2} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)}{\color{blue}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left|x\right|} \]
11. Step-by-step derivation

12. Applied rewrites 77.7%

    \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.5, x \cdot x, 1\right), x \cdot x, 1\right)}{\color{blue}{\sqrt{\pi}} \cdot \left|x\right|} \]
13. Add Preprocessing

Alternative 16: 52.2% accurate, 13.3× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(x, x, 1\right)}{\sqrt{\pi} \cdot \left|x\right|} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (fma x x 1.0) (* (sqrt PI) (fabs x))))

C

double code(double x) {
	return fma(x, x, 1.0) / (sqrt(((double) M_PI)) * fabs(x));
}

Julia

function code(x)
	return Float64(fma(x, x, 1.0) / Float64(sqrt(pi) * abs(x)))
end

Wolfram

code[x_] := N[(N[(x * x + 1.0), $MachinePrecision] / N[(N[Sqrt[Pi], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]
2. Add Preprocessing

4. Applied rewrites 100.0%

   \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \color{blue}{\left(\frac{\frac{0.5}{x \cdot x} + 1}{\left|x\right|} + \left(\frac{1.875}{\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left|x\right|\right) \cdot \left(x \cdot x\right)} + \frac{0.75}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left|x\right|}\right)\right)} \]

5. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \frac{e^{{\left(\left|x\right|\right)}^{2}}}{\left|x\right|}} \]
6. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot e^{{\left(\left|x\right|\right)}^{2}}}{\left|x\right|}} \]

   2. *-commutative N/A

      \[\leadsto \frac{\color{blue}{e^{{\left(\left|x\right|\right)}^{2}} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}}}{\left|x\right|} \]

   3. associate-/l* N/A

      \[\leadsto \color{blue}{e^{{\left(\left|x\right|\right)}^{2}} \cdot \frac{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}}}{\left|x\right|}} \]

   4. *-commutative N/A

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}}}{\left|x\right|} \cdot e^{{\left(\left|x\right|\right)}^{2}}} \]

   5. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}}}{\left|x\right|} \cdot e^{{\left(\left|x\right|\right)}^{2}}} \]

   6. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}}}{\left|x\right|}} \cdot e^{{\left(\left|x\right|\right)}^{2}} \]

   7. lower-sqrt.f64 N/A

      \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}}}}{\left|x\right|} \cdot e^{{\left(\left|x\right|\right)}^{2}} \]

   8. lower-/.f64 N/A

      \[\leadsto \frac{\sqrt{\color{blue}{\frac{1}{\mathsf{PI}\left(\right)}}}}{\left|x\right|} \cdot e^{{\left(\left|x\right|\right)}^{2}} \]

   9. lower-PI.f64 N/A

      \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{\mathsf{PI}\left(\right)}}}}{\left|x\right|} \cdot e^{{\left(\left|x\right|\right)}^{2}} \]

   10. lower-fabs.f64 N/A

       \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}}}{\color{blue}{\left|x\right|}} \cdot e^{{\left(\left|x\right|\right)}^{2}} \]

   11. unpow2 N/A

       \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}}}{\left|x\right|} \cdot e^{\color{blue}{\left|x\right| \cdot \left|x\right|}} \]

   12. sqr-abs N/A

       \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}}}{\left|x\right|} \cdot e^{\color{blue}{x \cdot x}} \]

   13. unpow2 N/A

       \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}}}{\left|x\right|} \cdot e^{\color{blue}{{x}^{2}}} \]

   14. lower-exp.f64 N/A

       \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}}}{\left|x\right|} \cdot \color{blue}{e^{{x}^{2}}} \]

   15. unpow2 N/A

       \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}}}{\left|x\right|} \cdot e^{\color{blue}{x \cdot x}} \]

   16. lower-*.f64 98.8

       \[\leadsto \frac{\sqrt{\frac{1}{\pi}}}{\left|x\right|} \cdot e^{\color{blue}{x \cdot x}} \]

7. Applied rewrites 98.8%

   \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\pi}}}{\left|x\right|} \cdot e^{x \cdot x}} \]
8. Step-by-step derivation

9. Applied rewrites 98.8%

   \[\leadsto \color{blue}{\frac{e^{x \cdot x}}{\sqrt{\pi} \cdot \left|x\right|}} \]

10. Taylor expanded in x around 0

    \[\leadsto \frac{1 + {x}^{2}}{\color{blue}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left|x\right|} \]
11. Step-by-step derivation

12. Applied rewrites 51.7%

    \[\leadsto \frac{\mathsf{fma}\left(x, x, 1\right)}{\color{blue}{\sqrt{\pi}} \cdot \left|x\right|} \]
13. Add Preprocessing

Alternative 17: 2.3% accurate, 16.1× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\sqrt{\pi} \cdot \left|x\right|} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 1.0 (* (sqrt PI) (fabs x))))

C

double code(double x) {
	return 1.0 / (sqrt(((double) M_PI)) * fabs(x));
}

Java

public static double code(double x) {
	return 1.0 / (Math.sqrt(Math.PI) * Math.abs(x));
}

Python

def code(x):
	return 1.0 / (math.sqrt(math.pi) * math.fabs(x))

Julia

function code(x)
	return Float64(1.0 / Float64(sqrt(pi) * abs(x)))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 / (sqrt(pi) * abs(x));
end

Wolfram

code[x_] := N[(1.0 / N[(N[Sqrt[Pi], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]
2. Add Preprocessing

4. Applied rewrites 100.0%

   \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \color{blue}{\left(\frac{\frac{0.5}{x \cdot x} + 1}{\left|x\right|} + \left(\frac{1.875}{\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left|x\right|\right) \cdot \left(x \cdot x\right)} + \frac{0.75}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left|x\right|}\right)\right)} \]

5. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \frac{e^{{\left(\left|x\right|\right)}^{2}}}{\left|x\right|}} \]
6. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot e^{{\left(\left|x\right|\right)}^{2}}}{\left|x\right|}} \]

   2. *-commutative N/A

      \[\leadsto \frac{\color{blue}{e^{{\left(\left|x\right|\right)}^{2}} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}}}{\left|x\right|} \]

   3. associate-/l* N/A

      \[\leadsto \color{blue}{e^{{\left(\left|x\right|\right)}^{2}} \cdot \frac{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}}}{\left|x\right|}} \]

   4. *-commutative N/A

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}}}{\left|x\right|} \cdot e^{{\left(\left|x\right|\right)}^{2}}} \]

   5. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}}}{\left|x\right|} \cdot e^{{\left(\left|x\right|\right)}^{2}}} \]

   6. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}}}{\left|x\right|}} \cdot e^{{\left(\left|x\right|\right)}^{2}} \]

   7. lower-sqrt.f64 N/A

      \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}}}}{\left|x\right|} \cdot e^{{\left(\left|x\right|\right)}^{2}} \]

   8. lower-/.f64 N/A

      \[\leadsto \frac{\sqrt{\color{blue}{\frac{1}{\mathsf{PI}\left(\right)}}}}{\left|x\right|} \cdot e^{{\left(\left|x\right|\right)}^{2}} \]

   9. lower-PI.f64 N/A

      \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{\mathsf{PI}\left(\right)}}}}{\left|x\right|} \cdot e^{{\left(\left|x\right|\right)}^{2}} \]

   10. lower-fabs.f64 N/A

       \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}}}{\color{blue}{\left|x\right|}} \cdot e^{{\left(\left|x\right|\right)}^{2}} \]

   11. unpow2 N/A

       \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}}}{\left|x\right|} \cdot e^{\color{blue}{\left|x\right| \cdot \left|x\right|}} \]

   12. sqr-abs N/A

       \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}}}{\left|x\right|} \cdot e^{\color{blue}{x \cdot x}} \]

   13. unpow2 N/A

       \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}}}{\left|x\right|} \cdot e^{\color{blue}{{x}^{2}}} \]

   14. lower-exp.f64 N/A

       \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}}}{\left|x\right|} \cdot \color{blue}{e^{{x}^{2}}} \]

   15. unpow2 N/A

       \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}}}{\left|x\right|} \cdot e^{\color{blue}{x \cdot x}} \]

   16. lower-*.f64 98.8

       \[\leadsto \frac{\sqrt{\frac{1}{\pi}}}{\left|x\right|} \cdot e^{\color{blue}{x \cdot x}} \]

7. Applied rewrites 98.8%

   \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\pi}}}{\left|x\right|} \cdot e^{x \cdot x}} \]
8. Step-by-step derivation

9. Applied rewrites 98.8%

   \[\leadsto \color{blue}{\frac{e^{x \cdot x}}{\sqrt{\pi} \cdot \left|x\right|}} \]

10. Taylor expanded in x around 0

    \[\leadsto \frac{1}{\color{blue}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left|x\right|} \]
11. Step-by-step derivation

12. Applied rewrites 2.4%

    \[\leadsto \frac{1}{\color{blue}{\sqrt{\pi}} \cdot \left|x\right|} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024240 
(FPCore (x)
  :name "Jmat.Real.erfi, branch x greater than or equal to 5"
  :precision binary64
  :pre (>= x 0.5)
  (* (* (/ 1.0 (sqrt PI)) (exp (* (fabs x) (fabs x)))) (+ (+ (+ (/ 1.0 (fabs x)) (* (/ 1.0 2.0) (* (* (/ 1.0 (fabs x)) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))))) (* (/ 3.0 4.0) (* (* (* (* (/ 1.0 (fabs x)) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))))) (* (/ 15.0 8.0) (* (* (* (* (* (* (/ 1.0 (fabs x)) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))) (/ 1.0 (fabs x)))))))